Dirichlet Forms and Symmetric Markov Processes [2nd rev. and ext. ed.] 9783110218091, 9783110218084

Citation preview

De Gruyter Studies in Mathematics 19 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Masatoshi Fukushima Yoichi Oshima Masayoshi Takeda

Dirichlet Forms and Symmetric Markov Processes Second revised and extended edition

De Gruyter

Mathematics Subject Classification 2010: 31-02, 60-02; 31C25, 60F10, 60J25, 60J45, 60J55.

ISBN 978-3-11-021808-4 e-ISBN 978-3-11-021809-1 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data Fukushima, Masatoshi 1935⫺ Dirichlet forms and symmetric Markov processes / by Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda. p. cm. ⫺ (De Gruyter studies in mathematics ; 19) Includes bibliographical references and index. ISBN 978-3-11-021808-4 (alk. paper) 1. Markov processes. 2. Dirichlet forms. I. Oshima, Yoichi. II. Takeda, Masayoshi. III. Title. QA274.7.F845 2010 519.21.33⫺dc22 2010041939

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

Part I of this book contains an introductory and comprehensive account of the theory of (symmetric) Dirichlet forms – an axiomatic extension of the classical Dirichlet integrals in the direction of Markovian semigroups. In Part II, this analytic theory is unified with the probabilistic potential theory based on symmetric Markov processes and developed further in conjunction with the stochastic analysis based on the additive functionals. We intend to organize it as a self-contained text book. Part I requires only a first course of functional analysis, while Part II can be read through with the help of “An introduction to Hunt processes” and “A summary on martingale additive functionals” being provided in Appendix. A brief summary at the beginning of each chapter, simple examples presented in many sections and the bibliographic notes stated at the end of the volume will serve to facilitate your use of the text. This is an outgrowth of Fukushima’s book “Dirichlet forms and Markov processes” published from Kodansha and North Holland in 1980, partly combined with Oshima’s lecture note “Lectures on Dirichlet spaces” delivered at Universität Erlangen-Nürnberg in 1988. Most ingredients in Fukushima’s book are maintained as the skeleton of the present volume. But they are reorganized and integrated with many new basic materials developed in the last decade. Some more of basic examples in finite dimensions are included. In the main text except for the first chapter and the last section, the underlying space is assumed to be locally compact. However the infinite dimensional non-locally compact situations can be handled in the present framework as well by making use of “Regular representations of Dirichlet spaces” being provided in the Appendix. We would like to express our hearty thanks to Professor H. Bauer for his warm encouragement in our joint writing of the book. We are grateful to Professors H. Ôkura and M. Tomisaki for their valuable comments on our preliminary drafts. We also thank Mrs. M. Tsukamoto for her great help in our preparation of the Tex file manuscript. Thanks are due to Dr. M. Karbe of Walter de Gruyter & Co. for his constant and truly generous cooperation. Osaka and Kumamoto, December 1993

Masatoshi Fukushima Yoichi Oshima Masayoshi Takeda

Preface to the second edition

We are pleased that our book has attracted constant interests of readers since its publication in 1994. A year ago, Professor Niels Jacob of Swansea University kindly conveyed us a generous offer from De Gruyter for us to prepare a second edition of the book. We are very grateful to them. In this second edition, §7.3 of the first edition is removed, while the following sections are newly added to the first one: §2.4 Capacities and Sobolev type inequalities §4.7 Irreducible recurrence and ergodicity §4.8 Recurrence and Poincaré type inequalities §6.4 Donsker–Varadhan type large deviation principle Otherwise the content in the first edition is maintained with minor modifications. More details about the changes will be stated at the end of Notes. In order to make the second edition more suitable to be a text book, Problems in the first editions are converted into Exercises together with many newly added ones and Solutions to them are provided at the end of the volume. We are grateful to Simon Albroscheit of De Gruyter for his encouragement and his kind cooperation in preparing the second edition. We sincerely hope that our book along with newly added materials will continue to serve as a text and a useful source of study on the related subjects. Osaka, Kumamoto and Sendai, July 2010

Masatoshi Fukushima Yoichi Oshima Masayoshi Takeda

Contents

Preface to the first and second edition

v

Notation

ix

I Dirichlet Forms

1

1 Basic theory of Dirichlet forms 1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Closed forms and semigroups . . . . . . . . . . . . . . . . . 1.4 Dirichlet forms and Markovian semigroups . . . . . . . . . 1.5 Transience of Dirichlet spaces and extended Dirichlet spaces 1.6 Global properties of Markovian semigroups . . . . . . . . . 2 Potential theory for Dirichlet forms 2.1 Capacity and quasi continuity . . . . . . . 2.2 Measures of finite energy integrals . . . . 2.3 Reduced functions and spectral synthesis . 2.4 Capacities and Sobolev type inequalities .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

3 3 6 16 25 37 53

. . . .

66 66 77 95 101

3 The scope of Dirichlet forms 109 3.1 Closability and the smallest closed extensions . . . . . . . . . . . 109 3.2 Formulae of Beurling–Deny and LeJan . . . . . . . . . . . . . . . 120 3.3 Maximum Markovian extensions . . . . . . . . . . . . . . . . . . 130

II Symmetric Markov processes

149

4 Analysis by symmetric Hunt processes 4.1 Smallness of sets and symmetry . . . . . . . . 4.2 Identification of potential theoretic notions . . . 4.3 Orthogonal projections and hitting distributions 4.4 Parts of forms and processes . . . . . . . . . . 4.5 Continuity, killing, and jumps of sample paths .

151 152 160 168 172 178

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

viii

Contents

4.6 4.7 4.8

Quasi notions, fine notions and global properties . . . . . . . . . . 189 Irreducible recurrence and ergodicity . . . . . . . . . . . . . . . . 201 Recurrence and Poincaré type inequalities . . . . . . . . . . . . . 207

5 Stochastic analysis by additive functionals 5.1 Positive continuous additive functionals and smooth measures 5.2 Decomposition of additive functionals of finite energy . . . . . 5.3 Martingale additive functionals and Beurling–Deny formulae . 5.4 Continuous additive functionals of zero energy . . . . . . . . 5.5 Extensions to additive functionals locally of finite energy . . . 5.6 Martingale additive functionals of finite energy and stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Forward and backward martingale additive functionals . . . .

. . . . .

. . . . .

221 222 241 256 261 270

. . 286 . . 295

6 Transformations of forms and processes 6.1 Perturbed Dirichlet forms and killing by additive functionals . . . 6.2 Traces of Dirichlet forms and time changes by additive functionals 6.2.1 Transient case . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 General case . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Transformations by supermartingale multiplicative functionals . . 6.4 Donsker–Varadhan type large deviation principle . . . . . . . . .

307 308 314 316 321 332 346

7 Construction of symmetric Markov processes 369 7.1 Construction of a Markovian transition function . . . . . . . . . . 369 7.2 Construction of a symmetric Hunt process . . . . . . . . . . . . . 373 Appendix A.1 Choquet capacities . . . . . . . . . . . . . . A.2 An introduction to Hunt processes . . . . . . A.3 A summary on martingale additive functionals A.4 Regular representations of Dirichlet spaces . A.5 Solutions to Exercises . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

382 382 384 406 422 439

Notes

453

Bibliography

461

Index

485

Notation

We use the following notations for a given measurable space .X; B/: f 2 B; f is an extended real valued function on X which is B-measurable f 2 B C ; f 2 B and f is non-negative f 2 Bb ; f 2 B and f is bounded. Given a function space A, we use the following: AC or AC ; the space of non-negative functions in A Ab ; the space of bounded functions in A. When X is a topological space, B.X/ (resp. B  .X/) denotes the family of all Borel measurable (resp. universally measurable, namely, measurable with respect to every probability measure on X ) subsets of X. C.X/ denotes the family of all real valued continuous functions on X. In this case, we write C C .X/; Cb .X/; f 2 B C .X/; f 2 Bb .X/ for C.X/C ; C.X/b ; f 2 B.X/C ; f 2 B.X/b , respectively. Further notations: D.u; v/; Dirichlet integral F ; Dirichlet space, i.e., the domain DŒE of a Dirichlet form E Fe ; extended Dirichlet space ¹F t º t 2Œ0;1 ; minimum completed admissible filtration P .ƒj†/; conditional probability with respect to a -field † S ; smooth measures S1 ; smooth measures in the strict sense S0 ; positive Radon measures of finite energy integrals S00 D ¹ 2 S0 W .X/ < 1; kU1 k1 < 1º AC c ; positive continuous additive functionals AC c1 ; positive continuous additive functionals in the strict sense.

Part I

Dirichlet Forms

Chapter 1

Basic theory of Dirichlet forms

The Dirichlet form on an L2 -space is defined as a Markovian closed symmetric form (§1.1). The link connecting the theory of Dirichlet forms with Markov processes is in that the Markovian nature of a closed symmetric form is equivalent to the Markovian properties of the associated semigroup and resolvent on L2 (§1.4). The domain of a Dirichlet form can be enlarged to the extended Dirichlet space. The recurrence and transience of a Markovian semigroup can be characterized in terms of the associated Dirichlet form and extended Dirichlet space (§1.5, §1.6). Thus the Dirichlet space in the original sense of Beurling and Deny is a specific transient extended Dirichlet space in the present context. Translation invariant Dirichlet forms, Sobolev spaces of order 1 and their extended Dirichlet spaces are studied in examples of §1.2, §1.4, §1.5 and §1.6.

1.1

Basic notions

Let H be a real Hilbert space with inner product .  ;  /. A non-negative definite symmetric bilinear form densely defined on H is henceforth called simply a symmetric form on H . To be precise, E is called a symmetric form on H if the following conditions are satisfied: (E.1) E is defined on DŒE  DŒE with values in R1 ; DŒE being a dense linear subspace of H , (E.2) E.u; v/ D E.v; u/; E.u C v; w/ D E.u; w/ C E.v; w/; aE.u; v/ D E.au; v/; E.u; u/  0; u; v; w 2 DŒE; a 2 R1 : We call DŒE the domain of E. The inner product .  ;  / on H is a specific symmetric form defined on the whole space H . Given a symmetric form E on H , ´ E˛ .u; v/ D E.u; v/ C ˛.u; v/; u; v 2 DŒE .1:1:1/ DŒE D DŒE defines a new symmetric form on H for each ˛ > 0: Note that the space DŒE is then a pre-Hilbert space with inner product E˛ . Furthermore E˛ and Eˇ determine equivalent metrics on DŒE for different ˛; ˇ > 0: If DŒE is complete with respect

4

1 Basic theory of Dirichlet forms

to this metric, then E is said to be closed. In other words, a symmetric form E is said to be closed if (E.3) un 2 DŒE; E1 .un  um ; un  um / ! 0; n; m ! 1 ) 9u 2 DŒE; E1 .un  u; un  u/ ! 0; n ! 1: Clearly DŒE is then a real Hilbert space with inner product E˛ for each ˛ > 0. We say that a symmetric form E is closable if the following condition is fulfilled: un 2 DŒE; E.un  um ; un  um / ! 0; n; m ! 1; .un ; un / ! 0; n ! 1 ) E.un ; un / ! 0; n ! 1:

.1:1:2/

Given two symmetric forms E .1/ and E .2/ ; E .2/ is said to be an extension of E .1/ if DŒE .1/   DŒE .2/  and E .2/ D E .1/ on DŒE .1/   DŒE .1/ . A necessary and sufficient condition for a symmetric form to possess a closed extension is that the symmetric form is closable. To see this, suppose the E is a closable symmetric form. Denote by A the set of all E1 -Cauchy sequences. Two sequences ¹un º; ¹u0n º 2 A are regarded to be equivalent if E1 .un  u0n ; un  u0n / ! 0; n ! 1: Denote by DŒE the set of all equivalence classes. This procedure provides us with a closed symmetric form E on H which is indeed the smallest closed extension of E. Exercise 1.1.1. Give the proof of the last sentence. Exercise 1.1.2. Show that the following is a sufficient condition for a symmetric form E to be closable: un 2 DŒE; .un ; un / ! 0; n ! 1 ) E.un ; v/ ! 0; n ! 1; 8v 2 DŒE:

.1:1:3/

We now consider a -finite measure space .X; B; m/ and take as a real Hilbert space H the L2 -space L2 .XI m/ consisting of square integrable m-measurable extended real valued functions on X. L2 .XI m/ is endowed with the inner product Z u.x/v.x/m.dx/; u; v 2 L2 .XI m/: .1:1:4/ .u; v/ D X

Let us call a symmetric form E on L2 .XI m/ Markovian symmetric form if the following property holds: (E:4) For each " > 0, there exists a real function " .t /; t 2 R1 ; such that " .t/ D t; 8t 2 Œ0; 1; "  " .t /  1 C "; 8t 2 R1 ; 0  " .t 0 /  " .t/  t 0  t whenever t < t 0 ; u 2 DŒE ) " .u/ 2 DŒE;

E." .u/; " .u//  E.u; u/:

and .1:1:5/ .1:1:6/

1.1

Basic notions

5

A Dirichlet form is by definition a symmetric form on L2 .XI m/ which is not only Markovian but also closed. This notion plays a primary role throughout the present volume. We now state two other conditions which look stronger but simpler than the Markovian condition .E:4/. Given a symmetric form E on L2 .XI m/, we say that the unit contraction (resp. every normal contraction) operates on E if the following .E:4/0 (resp. .E:4/00 ) is satisfied: (E.4)0 u 2 DŒE; v D .0 _ u/ ^ 1 ) v 2 DŒE; E.v; v/  E.u; u/: (E.4)00 u 2 DŒE; v is a normal contraction of u ) v 2 DŒE; E.v; v/  E.u; u/. Here a function v is called a normal contraction of a function u if jv.x/  v.y/j  ju.x/  u.y/j;

8x; y 2 X;

jv.x/j  ju.x/j;

8x 2 X:

We call v 2 L2 .XI m/ a normal contraction of u 2 L2 .XI m/ if some Borel version of v is a normal contraction of some Borel version of u. Notice the obvious implication: .E:4/00 ) .E:4/0 ) .E:4/. It turns out in §1.4 that these three conditions are actually equivalent if the form E is closed. Therefore we could have employed the simpler condition .E:4/0 or .E:4/00 instead of .E:4/ in the above definition of the Dirichlet form. However, as we shall see in Chapter 3, it is both practically and theoretically important to consider Markovian symmetric forms which are closable but not necessarily closed. They do not satisfy .E:4/0 nor .E:4/00 in general. After Chapter 2 throughout the main text except for the last section §7.3, we shall make a topological assumption that X is a locally compact separable metric space m is a positive Radon measure on X such that suppŒm D X

.1:1:7/

i.e., m is a non-negative Borel measure on X finite on compact sets and strictly positive on non-empty open sets.1 We then denote by C.X/ the space of all real continuous functions on X and consider its subspaces Cb .X/ D ¹u 2 C.X/ W u is boundedº; C0 .X/ D ¹u 2 C.X/ W suppŒu is compactº; C1 .X/ D ¹u 2 C.X/ W 8" > 0; 9K compact; ju.x/j < "; 8x 2 X n Kº: 1 We shall occasionally indicate the possibilities of replacing the topological assumption (1.1.7) by less restrictive ones.

6

1 Basic theory of Dirichlet forms

The functions in C1 .X/ are said to be vanishing at infinity. Since any u 2 C.X/ vanishing m-a.e. on X is identically zero, the subspace L2 .XI m/\C.X/ of C.X/ can be viewed as a subspace of L2 .XI m/. A core of a symmetric form E is by definition a subset C of DŒE \ C0 .X/ such that C is dense in DŒE with E1 -norm and dense in C0 .X/ with uniform norm. E is called regular if (E.5) E possesses a core. It is clear that E is regular if and only if the space DŒE \ C0 .X/ is a core of E. A core C of E is said to be standard if (C.1) C is a dense linear subspace of C0 .X/. For any " > 0, there exists a real function " .t/ satisfying (1.1.5) such that " .u/ 2 C whenever u 2 C. By a special standard core, we mean a standard core C of E satisfying additionally (C.2) C is a dense subalgebra of C0 .X/. For any compact set K and relatively compact open set G with K  G, C admits an element u such that u  0; u D 1 on K and u D 0 on X n G. Suppose that E is a regular Dirichlet form. Then DŒE \ C0 .X/ is a standard core. We shall see in §1.4 that it is special standard as well. For an m-measurable function u, the support suppŒu  m of the measure u  d m is simply denoted by suppŒu. If u 2 C.X/, then suppŒu is just the closure of ¹x 2 X W u.x/ ¤ 0º. We say that a symmetric form E possesses the local property or simply E is local if (E.6) u; v 2 DŒE, suppŒu and suppŒv are disjoint compact sets ) E.u; v/ D 0. E will be said to be strong local if (E.7) u; v 2 DŒE, suppŒu and suppŒv are compact, v is constant on a neighbourhood of suppŒu ) E.u; v/ D 0.

1.2 Examples First of all, we exhibit a typical example of a Markovian symmetric form which is not closed. Example 1.2.1. Let us consider the case where X is a domain D of the Euclidean d -space Rd : C01 .D/ denotes the set of all infinitely differentiable functions on D

1.2

Examples

7

with compact support in D. Then E.u; v/ D

d Z X i;j D1 D

@u.x/ @v.x/ ij .dx/ @xi @xj

Z

C

DDnd

.u.x/  u.y//.v.x/  v.y//J.dxdy/

Z C DŒE D

u.x/v.x/k.dx/;

(1.2.1)

D 1 C0 .D/

(1.2.2)

is a Markovian symmetric form on L2 .DI m/. Here m is a positive Radon measure on D with suppŒm D D. ij .1  i; j  d / are Radon measures on D such that for any  2 Rd and any compact set K  D, d X

i j ij .K/  0;

ij .K/ D j i .K/;

1  i; j  d:

.1:2:3/

i;j D1

J is a positive symmetric Radon measure on the product space D  D off the diagonal d such that for any compact set K and open set D1 with K  D1  D Z jx  yj2 J.dxdy/ < 1; J.K; D n D1 / < 1: .1:2:4/ KKnd

k is a positive Radon measure on D. The finiteness condition (1.2.4) of J is a necessary and sufficient condition that the second integral of (1.2.1) converges absolutely for any u; v 2 C01 .D/, as we easily see by dividing the domain D  D n d of integration into a sum of the sets D1  D1 n d; .D n D1 /  D1 ; D1  .D n D1 / and .D n D1 /  .D n D1 / n d; D1 being a relatively compact open set containing the supports of u and v. To see that the form E defined by (1.2.1) and (1.2.2) is a symmetric form on 2 L .DI m/, it suffices to show that the first term of the right-hand side of (1.2.1) is non-negative definite. Divide Rd into cubes with sides parallel to the axis and of side length ı > 0. Denote by ¹C 1 ; : : : ; C l º those cubes whose closures are .k/ / .k/ D i ; u contained in D. Take a point .k/ from each cube C k and put @u. @xi being any element of C01 .D/. Then by the Lebesgue theorem d Z X i;j D1 D

l.ı/  X d  X @u.x/ @u.x/ .k/ .k/ ij .dx/ D lim i j ij .C k / @xi @xj ı#0

which is non-negative in view of (1.2.3).

kD1

i;j D1

8

1 Basic theory of Dirichlet forms

In order to verify the Markovian property .E:4/, choose, for each " > 0, an infinitely differentiable function " .t/ satisfying the condition (1.1.5). Such function can easily be constructed (see Exercise 1.2.1 below). Then, for any u 2 C01 .D/; " .u/ 2 C01 .D/ and E." .u/; " .u// D

d Z X i;j D1 D

j"0 .u.x//j2

@u.x/ @u.x/ ij .dx/ @xi @xj

Z

C

DDnd

." .u.x//  " .u.y///2 J.dxdy/

Z C

" .u.x//2 k.dx/; D

which is not greater than E.u; u/ because we know from (1.1.5) the inequalities 0  "0 .t/  1; j" .t/  " .s/j  jt  sj and j" .t /j  jt j. Exercise 1.2.1. Consider a mollifier, e.g., j.x/ D exp.1=.1jxj2R// for jxj < 1 and j.x/ D 0 for jxj  1, where is a positive constant to make jxj1 j.x/dx D 1. We put jı .x/ D ı 1 j.ı 1 x/ for ı > 0. For any " > 0, we consider 1 Ra function " .t/ D .."/ _ t/ ^ .1 C "/ on R and put " .t / D jı  " .t / D R1 jı .t  s/ " .s/ds, 0 < ı < ". Show that " .t / is then an infinitely differentiable function satisfying (1.1.5). When u and v 2 C01 .D/ have disjoint supports, (1.2.1) reduces to E.u; v/ D R 2 DDnd u.x/v.y/J.dxdy/. We can conclude from this that the form (1.2.1) possesses the local property .E:6/ if and only if J D 0. The Markovian symmetric form (1.2.1), (1.2.2) is not closed since the domain is restricted to C01 .D/. However it is closable in many cases. Several conditions on the measures ij yielding the closability of the associated forms are collected in

3:1. We also prove in 3:2 that (1.2.1) is the most general expression of a closable Markovian symmetric form on L2 .DI m/ possessing C01 .D/ as its core. The next two examples concern the simplest cases that the measures ij in (1.2.1) are the Lebesgue measures up to constant factors. In these cases we can describe explicitly those domains larger than C01 .D/ which make the resulting forms closed. Let us denote by L2 .D/ the L2 -space with respect to the Lebesgue measure dx on D.

1.2

Examples

9

Example 1.2.2. Let I D .r1 ; r2 / be a one-dimensional interval: 1  r1 < r2  C1. Let m and k be positive Radon measures on I with suppŒm D I . We put Z r2 du.x/ dv.x/ dx (1.2.5) D.u; v/ D dx dx r1 F R D ¹u 2 L2 .I I m/ \ L2 .I I k/ W u is absolutely continuous and D.u; u/ < 1º: Then the form E defined by 8 R 0; j.x; X n U" .x// is, as a function of x 2 X, locally integrable with respect to m. Here U" .x/ is the "-neighbourhood of x. R R (j.2) X u.x/.j v/.x/m.dx/ D X .j u/.x/v.x/m.dx/ . 1/ for any u; v 2 B C .X/. j then determines a positive Radon measure J.dxdy/ on X  X n d .d is the diagonal set) by ¯ R R ®R X X nd f .x; y/J.dxdy/ D X X f .x; y/j.x; dy/ m.dx/; .1:2:22/ f 2 C0 .X  X n d /: From the condition (j.2) follows the symmetry of J : Z Z f .x; y/J.dxdy/ D f .y; x/J.dxdy/; X X nd

X X nd

f 2 C0 .X  X n d /:

Put ´ R E.u; v/ D X X nd .u.x/  u.y//.v.x/  v.y//J.dxdy/ DŒE D ¹u 2 L2 .XI m/ W u is Borel measurable; E.u; u/ < 1º:

.1:2:23/

.1:2:24/

Then E is a Dirichlet form on L2 .XI m/ provided that DŒE is dense in L2 .XI m/.

1.2

Examples

15

We first show that for a Borel function u, u D 0 m-a.e.

)

E.u; u/ D 0:

.1:2:25/

Suppose u D 0 m-a.e. Put K;" D ¹.x; y/ 2 K  K W .x; y/ > "º for " > 0 and K compact. Then Z

K;"

Z .u.x/  u.y//2 J.dxdy/  2 Z D4

K;"

K;"

.u.x/2 C u.y/2 /J.dxdy/

Z

u.x/2 J.dxdy/  4

K

u.x/2 J.x; X n U" .x//m.dx/ D 0:

By letting " # 0 and K " X, we get E.u; u/ D 0. We can see the Markovian property .E:4/ in the same way as in Example 1.2.1. But now we can directly check the stronger condition .E:4/00 : every normal contraction operates on the form (1.2.24). To prove the closedness, consider ul 2 DŒE such that E1 .ul  um ; ul  um / ! 0; l; m ! 1. Since ul converges in L2 , there are subsequence lk and a set N 2 B.X/ with m.N / D 0 such that ulk .x/ ulk .x/ D 0 on N . Then converges on X n N . Put e ulk .x/ D ulk .x/ on X n N and e e ulk .x/ has a limit u.x/ everywhere and ul converges to u in L2 . Moreover, E.u  um ; u  um / Z lim ¹.ulk .x/  ulk .y//  .um .x/  um .y//º2 J.dxdy/ D X X nd lk !1

 lim inf E.ulk  um ; ulk  um /: lk !1

The last term can be made arbitrarily small for sufficiently large m. Thus um is E1 -convergent to u 2 DŒE. When X is a domain D  Rd ; C01 .D/  DŒE if and only if R (j.3) KK jx  yj2 j.x; dy/m.dx/ < 1 for any compact K  D. Thus E is a Dirichlet form on L2 .DI m/ under the conditions (j.1), (j.2) and (j.3). Example 1.2.5. Let E be a countable set with discrete topology. Let Q D .qij /i;j 2E be a matrix satisfying (i) 0 < qi i < 1; i 2 E. (ii) qij  0; i; j 2 E; i ¤ j: P (iii) j 2E qij  0; i 2 E:

16

1 Basic theory of Dirichlet forms

A matrix Q possessing those three properties is said to be Q-matrix. We shall identify a measure m on E and a sequence ¹mi º by mi D m.¹iº/; i 2 E: Suppose there exists a measure m0 on E such that 0 < m0i < 1;

i 2 E;

m0i qij D mj0 qj i ;

i; j 2 E:

.1:2:26/

For real valued functions f; g on I; we let E.f; g/ D

X 1 X .f .i/  f .j //.g.i/  g.j //qij m0i C f .i/g.i/ki m0i ; 2 i;j 2E

i2E

.1:2:27/ whenever the right-hand side makes sense where ki D qi i  j ¤i qij ; i 2 E: Denote by C the set of real valued functions on E vanishing except on finite number of points. For f; g 2 C; E.f; g/ is finite and E.f; g/ D .Qf; g/L2 .E Im0 / : Take an arbitrary measure m on E with 0 < mi < 1 for any i 2 I; and let P

F R D ¹f 2 L2 .EI m/ W E.f; f / < 1º:

.1:2:28/

Then .E; F R / is a Dirichlet form on L2 .EI m/: Denote by F 0 the E1 -closure of C: .E; F 0 / is then a regular Dirichlet form on L2 .EI m/:

1.3

Closed forms and semigroups

In this section we consider only an abstract real Hilbert space H with inner product .  ;  /. Consider a family ¹T t ; t > 0º of linear operators on H satisfying the following conditions: (T t .1) Each T t is a symmetric operator with domain D.T t / D H . (T t .2) Semigroup property: T t Ts D T t Cs ; t; s > 0. (T t .3) Contraction property: .T t u; T t u/  .u; u/; t > 0; u 2 H . Then ¹T t ; t > 0º is called a semigroup (of symmetric operators) on H . It is called strongly continuous if in addition (T t .4) .T t u  u; T t u  u/ ! 0; t # 0; u 2 H . A resolvent on H is by definition a family ¹G˛ ; ˛ > 0º of linear operators on H satisfying the following conditions: (G˛ .1) Each G˛ is a symmetric operator with domain D.G˛ / D H . (G˛ .2) Resolvent equation: G˛  Gˇ C .˛  ˇ/G˛ Gˇ D 0: (G˛ .3) Contraction property: .˛G˛ u; ˛G˛ u/  .u; u/; ˛ > 0; u 2 H:

1.3

Closed forms and semigroups

17

If an additional condition (G˛ .4) .˛G˛ u  u; ˛G˛ u  u/ ! 0; ˛ ! 1; u 2 H is satisfied, the resolvent is said to be strongly continuous. Exercise 1.3.1. Given a strongly continuous semigroup ¹T t ; t > 0º on H , show that the Bochner integral Z 1 e ˛t T t udt .1:3:1/ G˛ u D 0

determines a strongly continuous resolvent ¹G˛ ; ˛ > 0º on H . This is called the resolvent of the given semigroup. The generator A of a strongly continuous semigroup ¹T t ; t > 0º on H is defined by

8 0º on H , let us assume that G˛ u D 0. Then Gˇ u D 0; 8ˇ > 0, from .G˛ :2/, and u D limˇ !1 ˇGˇ u D 0 from .G˛ :4/. Hence G˛ is invertible and we can set ´ Au D ˛u  G˛1 u .1:3:3/ D.A/ D G˛ .H /: This operator A is easily seen to be independent of ˛ > 0 and is called the generator of a given resolvent ¹G˛ ; ˛ > 0º. Lemma 1.3.1. (i) The generator of a strongly continuous resolvent is a nonpositive definite self-adjoint operator. (ii) The generator of a strongly continuous semigroup on H coincides with the generator of its resolvent. Proof. (i) Let ¹G˛ ; ˛ > 0º be a strongly continuous resolvent on H . Since G˛ is a symmetric operator defined on the whole space H , the inverse G˛1 and hence the generator A of G˛ are self-adjoint. Put f .˛/ D .u; G˛ u/ for u 2 H , then by the resolvent equation .d=d˛/f .˛/ D .G˛ u; G˛ u/  0. From .G˛ :3/; jf .˛/j  .1=˛/.u; u/ ! 0, ˛ ! 1. Therefore, f .˛/  0, that is, G˛ is non-negative definite. This implies the same property of A because, for u 2 D.A/; .Au; u/ D lim˛#0 .Au C ˛u; u/ D lim˛#0 .G˛1 u; u/  0.

18

1 Basic theory of Dirichlet forms

(ii) Given a strongly continuous semigroup ¹T t ; t > 0º on H , denote its resolvent by ¹G˛ ; ˛ > 0º. Consider the generators A and A0 of T t and G˛ respectively. Take u 2 D.A0 /, then u D G˛ v; 9v 2 H . From (1.3.1) 1 1 ˛t .e T t u  u/ D  t t

Z 0

t

e ˛s Ts vds ! v;

t # 0;

which means u 2 D.A/ and Au D ˛u  v D A0 u, proving A0  A. Conversely, take u 2 D.A/ and put lim t #0 .1=t/.e ˛t T t u  u/ D v; w D u  G˛ v. Then lim t #0 .1=t/.e ˛t T t w  w/ D 0 and hence .Aw; w/ C ˛.w; w/ D 0. Since .Aw; w/ D lim t #0 .1=t/¹.w; w/  .T t w; w/º is non-negative by .T t :3/, we get w D 0 proving A  A0 . By this lemma, we are led from semigroups to self-adjoint operators. In the following we make full use of the spectral calculus relevant to self-adjoint operators.8 A symmetric operator S on H satisfying D.S / D H , S 2 D S is called a projection operator. A family ¹E , 1 <  < 1º of projection operators on H is called a spectral family if E E D E ,   , lim0 # E0 u D E u, lim!1 E u D 0, lim!1 E u D u, u 2 H . Then 0  .E u; u/ " .u; u/,  " 1, for u 2 H . Moreover, .E u; v/ is of bounded variation in  for u; v 2 H . Given a spectral family ¹E ; 1 <  < 1º on H and a continuous function ./ on R1 , a self-adjoint operator RA on H can be uniquely defined by the 1 following formula. This A is denoted by 1 ./dE . 8 R 0º and ¹G˛ D .˛  A/1 ; ˛ > 0º are a strongly continuous semigroup and a strongly continuous resolvent on H respectively. (ii) The generator of T t of (i) coincides with A. The strongly continuous semigroup possessing A as its generator is unique. The same statement holds for the resolvent. Proof. (i) This R can be proved easily with the aid of the formula ..A/u, .A/v/ D Œ0;1/ ./ ./d.E u; v/, u 2 D..A//, v 2 D. .A// which holds for any Rcontinuous functions  and on Œ0; 1/. For instance, from ˛G˛  I D Œ0;1/ .˛=.˛ C /  1/dE follows .˛G˛ u  u; ˛G˛ u  u/ D R 2 Œ0;1/ ..˛=˛ C /  1/ d.E u; u/ ! 0, ˛ ! 1, u 2 H , the strong continuity of G˛ . (ii) We see for u 2 H Z Z 2 2  d.E G˛ u; G˛ u/ D d.E u; u/ < 1; 2 Œ0;1/ Œ0;1/ .˛ C / which means that G˛ .H /  D.A/. Then the obvious equalities .˛  A/G˛ u D u, u 2 H and G˛ .˛  A/u D u, u 2 D.A/, imply that the generator of ¹G˛ , ˛ > 0º is equal to A. On the other hand, G˛ is the resolvent of T t in the sense of (1.3.1). Hence A must also be the generator of T t in view of Lemma 1.3.1. Finally, let ¹G˛0 ; ˛ > 0º be any strongly continuous resolvent with generator A. Put w D G˛ u  G˛0 u; u 2 H , then .˛  A/w D 0. Since A is non-negative definite, we get w D 0, namely, G˛0 D G˛ . The uniqueness concerning the strongly continuous semigroup ¹T t ; t > 0º follows from the right continuity of .T t u; v/ in t .u; v 2 H / and the uniqueness theorem of the Laplace transform. The above two lemmas tell us that there are one to one correspondences as indicated in Diagram 1 among the family of non-positive definite self-adjoint operators on H , the family of strongly continuous semigroups, and the family of strongly continuous resolvents. exp.tA/  3 

+   ¹T t ; t > 0º 

A generator (1.3.1)

1 Q QQQ .˛  A/ k QQQ s Q -

(?) Diagram 1.

¹G˛ ; ˛ > 0º

20

1 Basic theory of Dirichlet forms

Exercise 1.3.2. Show that the correspondence (?) in Diagram 1 is given by T t u D lim e ˇ !1

tˇ

1 X .tˇ/n .ˇGˇ /n u; nŠ

u 2 H:

.1:3:5/

nD0

We now state the main theorem of this section. Theorem 1.3.1.9 There is a one to one correspondence between the family of closed symmetric forms E on H and the family of non-positive definite self-adjoint operators A on H . The correspondence is determined by 8 p 0º generated by A satisfies G˛ .H /  DŒE;

E˛ .G˛ u; v/ D .u; v/;

u 2 H; v 2 DŒE:

This follows easily from the expression 8 R 0 and u 2 H . We easily see that the family of linear operators ¹G˛ ; ˛ > 0º defined this way is a strongly continuous resolvent. For instance, the contraction property follows from ˛.G˛ u; G˛ u/  E˛ .G˛ u; G˛ u/ D .u; G˛ u/ and the Schwarz inequality. To see the strong continuity, it suffices to show the 9 Cf.

T. Kato [1].

1.3

Closed forms and semigroups

21

strong convergence ˇGˇ u ! u; ˇ ! 1, only for u 2 DŒE because DŒE is dense in H and ˇGˇ is contractive. For u 2 DŒE; ˇ.ˇGˇ u  u; ˇGˇ u  u/  Eˇ .ˇGˇ u  u; ˇGˇ u  u/ D ˇ 2 .Gˇ u; u/  ˇ.u; u/ C E.u; u/  E.u; u/, which implies the desired convergence. Denote by A the generator of this resolvent ¹G˛ ; ˛ > 0º. Since A is nonnegative definite and self-adjoint, we may associate with A a closed symmetric form E 0 by the formula (1.3.6). We claim that E 0 D E. From (1.3.7), G˛ .H /  DŒE 0  and E˛0 .G˛ u; G˛ v/ D .G˛ u; v/, which also equals E˛ .G˛ u; G˛ v/ by (1.3.9) for u; v 2 H . Thus E 0 D E on G˛ .H /  G˛ .H /. But the same equations (1.3.7) and (1.3.9) imply that G˛ .H / is dense both in DŒE 0  and in DŒE proving E 0 D E. (iii) For a given E, A satisfying (1.3.6) is unique, because the resolvent ¹G˛ ; ˛ > 0º generated by such A satisfies the equation (1.3.7), which in turn means that ¹G˛ ; ˛ > 0º and hence A are uniquely determined by E. In the above proof we actually showed that the correspondence between E and the resolvent of A is characterized by the equation (1.3.7). A restatement of this is the following. Corollary 1.3.1. The correspondence in Theorem 1:3:1 can be characterized by ´ D.A/  DŒE .1:3:10/ E.u; v/ D .Au; v/; u 2 D.A/; v 2 DŒE: Example 1.3.1. Let A0 be the self-adjoint operator on L2 .I I m/ corresponding to the Dirichlet form E of Example 1.2.2 with domain F 0 . D.A0 / D ¹u 2 L2 .I I m/ \ L2 .I I k/ W u is absolutely continuous, a version of du=dx is of bounded variation on each finite subinterval of I; .1=2/.d.du=dx/ ud k/ is absolutely continuous with respect to m and the derivative belongs to L2 .I I m/; u.ri / D 0 if ri is a regular boundaryº; (1.3.11) A0 u D

1 du 2 d dx

 ud k ; dm

u 2 D.A0 /:

(1.3.12)

We know from (1.3.10) that u 2 D.A/; Au D f 2 L2 .I I m/ if and only if u 2 DŒE; E.u; v/ D .f; v/; 8v 2 DŒE. Since C01 .I / is dense in DŒE D F 0 ,

22

1 Basic theory of Dirichlet forms

this is also equivalent to u 2 F 0 ; E.u; v/ D .f; v/; 8v 2 C01 .I /, from which we get the above expression of A0 . Exactly in the same way, we can describe the self-adjoint operator A0 on L2 .D/ corresponding to the Dirichlet form E of Example 1.2.3 with domain H01 .D/ as follows: 8 0º be the strongly continuous semigroup and resolvent corresponding to A. Then

1.3

Closed forms and semigroups

23

(i) T t .H /  DŒE; E.T t u; T t u/  .1=2t/¹.u; u/  .T t u; T t u/º  E.u; u/, u 2 DŒE. (ii) G˛ .H /  DŒE; E˛ .G˛ u; v/ D .u; v/, u 2 H , v 2 DŒE. (iii) The following convergence takes place strongly in DŒE for any u 2 DŒE: T t u ! u; t # 0; .1=t/.G1 u  e t G1 T t u/ D .1=t /.G1 u  e t T t G1 u/ ! u, t # 0, ˛G˛ u ! u, ˛ ! 1. Proof. (ii) has been shown using (1.3.8). Other assertions can be proved in the same way. By integrating e 2t   .1=2t /.1  e 2t  / with d.E u; u/, we get w t D .1=t/.G1 u  e t G1 T t u/, we have E1 .w t  u; w t  u/ D R (i). For t .C1/ /=t. C 1/  1/2 . C 1/d.E u; u/ ! 0, t # 0, u 2 DŒE.  Œ0;1/ ..1  e For a semigroup ¹T t ; t > 0º and a resolvent ¹G˛ ; ˛ > 0º on H , we define symmetric forms E .t / and E .ˇ / on H by 1 E .t / .u; v/ D .u  T t u; v/; t E .ˇ / .u; v/ D ˇ.u  ˇGˇ u; v/;

u; v 2 H; u; v 2 H:

(1.3.15) (1.3.16)

The next lemma justifies our saying that E .t / and E .ˇ / are approximating forms determined by T t and Gˇ respectively. Lemma 1.3.4. Consider E; A; T t and G˛ of Lemma 1:3:3 and let E .t / and E .ˇ / be the approximating forms determined by T t and G˛ respectively. (i) For u 2 H; E .t / .u; u/ is non-decreasing as t # 0 and ´ DŒE D ¹u 2 H W lim t !0 E .t / .u; u/ < 1º E.u; v/ D lim t !0 E .t / .u; v/; u; v 2 DŒE: (ii) For any u 2 H; E .ˇ / .u; u/ is non-decreasing as ˇ " 1 and ´ DŒE D ¹u 2 H W limˇ !1 E .ˇ / .u; u/ < 1º E.u; v/ D limˇ !1 E .ˇ / .u; v/; u; v 2 DŒE:

.1:3:17/

.1:3:18/

This lemma can be proved using the spectral family just as in the proof of the preceding lemma. This lemma is very useful in that it provides us with a simple direct description of E in terms of T t and G˛ . In particular, a direct correspondence between the family of closed symmetric forms and the family of strongly continuous resolvents is given by (1.3.18) and (1.3.7).

24

1 Basic theory of Dirichlet forms

Such a correspondence can be extended to the relationship between resolvents which are not necessarily strongly continuous and closed forms whose domains are not necessarily dense in H . We call E a symmetric form on H in the wide sense if E satisfies all conditions of the symmetric form except for the denseness of DŒE in H . Theorem 1.3.2. (i) Given a resolvent ¹G˛ ; ˛ > 0º on H not necessarily strongly continuous, let E .ˇ / the approximating form determined by it. Then E .ˇ / .u; u/ is non-decreasing as ˇ " 1 for each u 2 H . Furthermore, the form defined by the right-hand side of .1:3:18/ is a closed symmetric form on H in the wide sense. (ii) Conversely, given a closed symmetric form E on H in the wide sense, .1:3:7/ defines a resolvent ¹G˛ ; ˛ > 0º on H not necessarily strongly continuous. (iii) The correspondence given in (i) and in (ii) are reciprocal each other. Proof. (i) The resolvent equation and the contraction property of G˛ imply that G˛ is non-negative definite and .Gˇ u; u/  .1=ˇ/.u; u/. Hence E .ˇ / .u; u/  0. By the resolvent equation again, 8 d < dˇ E .ˇ / .u; u/ D .ˇGˇ u  u; ˇGˇ u  u/  0 :

d 2 .ˇ / E .u; u/ dˇ 2

D 2.v; Gˇ v/  0;

.1:3:19/

where v D ˇGˇ u  u. We see in particular that E .ˇ / .u; u/ is non-decreasing as ˇ " 1. Therefore, we can define a symmetric form E on H in the wide sense by the right-hand side of (1.3.18). In order to see that E is closed, assume that un 2 DŒE constitutes a Cauchy sequence with respect to E1 . Then un converges to some element u 2 H in H . We have for each ˇ > 0 E .ˇ / .un  u; un  u/ D limm!1 E .ˇ / .un  um ; un  um /  limm!1 E.un  um ; un  um /, which can be made arbitrarily small uniformly in ˇ for sufficiently large n. After letting ˇ tend toward 1, we see that u 2 DŒE and limn!1 E.un  u; un  u/ D 0. We note that .ˇGˇ u  u; ˇGˇ u  u/ ! 0; ˇ ! 1; lim E˛.ˇ / .u; v/ D E˛ .u; v/;

ˇ !1

8u 2 DŒE

8˛ > 0; 8u; v 2 DŒE;

(1.3.20) (1.3.21)

1.4 Dirichlet forms and Markovian semigroups

25

.ˇ /

where E˛ .u; v/ D ˇ.u  ˇGˇ C˛ u; v/. (1.3.20) is a consequence of (1.3.19). (1.3.21) follows from E˛.ˇ / .u; v/  E .ˇ / .u; v/ D ˇ 2 ..Gˇ  Gˇ C˛ /u; v/ D ˛ˇ 2 .Gˇ C˛ Gˇ u; v/ D ˛.ˇGˇ u; ˇGˇ C˛ v/; which converges to ˛.u; v/ by virtue of (1.3.20). The present ¹G˛ ; ˛ > 0º and E are related by (1.3.7). In fact, we have, for u 2 H, E˛.ˇ / .G˛ u; G˛ u/ D ˇ.Gˇ C˛ u; G˛ u/ D ˇ.u; Gˇ C˛ G˛ u/ D .u; G˛ u/  .u; Gˇ C˛ u/ ! .u; G˛ u/;

ˇ ! 1;

and hence G˛ u 2 DŒE and E˛ .G˛ u; G˛ u/ D .u; G˛ u/ in view of (1.3.21). Furthermore, (1.3.20) implies that E˛ .G˛ u; v/ D lim E˛.ˇ / .G˛ u; v/ D lim ˇ.Gˇ C˛ u; v/ ˇ !1

ˇ !1

D lim .u; ˇGˇ C˛ v/ D .u; v/ ˇ !1

for v 2 DŒE. (ii) The proof is similar to the proof (ii) of Theorem 1.3.1. (iii) This is clear because both relations are characterized by (1.3.7).

1.4

Dirichlet forms and Markovian semigroups

Consider a -finite measure space .X; B; m/. A linear operator S on L2 .XI m/ with D.S / D L2 .XI m/ is called Markovian if 0  S u  1 m-a.e. whenever u 2 L2 .XI m/; 0  u  1 m-a.e. We say that S is positivity preserving if S u  0 m-a.e. whenever u 2 L2 .XI m/ and u  0 m-a.e. Theorem 1.4.1. Let E be a closed symmetric form on L2 .XI m/. Let ¹T t ; t > 0º and ¹G˛ ; ˛ > 0º be the strongly continuous semigroup and the strongly continuous resolvent on L2 .XI m/ which are associated with E in the manner of the preceding section. Then the next five conditions are equivalent to each other: (a) T t is Markovian for each t > 0. (b) ˛G˛ is Markovian for each ˛ > 0. (c) E is Markovian. (d) The unit contraction operates on E. (e) Every normal contraction operates on E.

26

1 Basic theory of Dirichlet forms

A semigroup (resp. a resolvent) on L2 .XI m/ satisfying condition (a) (resp. (b)) is called a Markovian semigroup (resp. a Markovian resolvent). In particular, Theorem 1.4.1 means that the family of all Dirichlet forms on L2 .XI m/ and the family of all strongly continuous Markovian semigroups on L2 .XI m/ stand in one to one correspondence. The implications (a))(b) and (b))(a) are evident from (1.3.1) and (1.3.5) respectively. (e))(d))(c) is trivial. Hence it suffices to prove the relations (c))(b) and (b))(e). Proof of (c))(b). Fix ˛ > 0 and u 2 L2 .XI m/ such that 0  u  1 m-a.e. Introduce a quadratic form on DŒE by  u u ; v 2 DŒE: .1:4:1/ .v/ D E.v; v/ C ˛ v  ; v  ˛ ˛ Then, by virtue of (1.3.7), we have .G˛ u/ C E˛ .G˛ u  v; G˛ u  v/ D

.v/;

v 2 DŒE:

.1:4:2/

In other words, G˛ u is the unique element of DŒE minimizing . Now suppose that E is Markovian, i.e., there exists for each " > 0 a function " .t / satisfying the Markovian condition (E.4). Put e  " .t/ D .1=˛/˛" .˛t / and w D e  " .G˛ u/. Then w 2 DŒE;

E.w; w/  E.G˛ u; G˛ u/:

.1:4:3/

On the other hand, je  " .t/  sj  jt  sj for s 2 Œ0; 1=˛ and t 2 R1 and so jw.x/  u.x/=˛j  jG˛ u.x/  u.x/=˛j m-a.e. Hence .w  u=˛; w  u=˛/  .G˛ u  u=˛; G˛ u  u=˛/. Combining this with (1.4.3), we get .w/  .G˛ u/ which implies that w D G˛ u. In particular "  G˛ u  1=˛ C ". Since " is arbitrary, the Markovian property of ˛G˛ is proven. The implication (b))(e) will be proved under the topological assumption (1.1.7) on .X; m/.10 We prepare a lemma. Lemma 1.4.1. Assume .1:1:7/.11 (i) If S is a positive symmetric linear operator on L2 .XI m/, then there exists a unique positive symmetric Radon measure  on the product space X  X satisfying the following property: for any Borel functions u; v 2 L2 .XI m/ Z u.x/v.y/.dx; dy/: .1:4:4/ .u; S v/ D X X

10 Actually this implication and consequently Theorem 1.4.1 can be proved without any topological assumption. We refer the readers to recent books by Z. Ma and M. Röckner [1] and by N. Bouleau and F. Hirsch [2] in this connection. 11 See C. Dellacherie and P. A. Meyer [2; p. 80] for the proof of this lemma under a less restrictive topological assumption than (1.1.7).

1.4 Dirichlet forms and Markovian semigroups

27

(ii) If in addition S is Markovian, .X  E/  m.E/;

8E 2 B.X/:

.1:4:5/

Proof. (ii) follows from (i). To prove (i), let us consider a function f .x; y/ D Pl iD1 ui .x/vi .y/; ui ; vi 2 C0 .X/, on the space X  X and let l X I.f / D .ui ; S vi /:

.1:4:6/

iD1

Assuming that f .x;y/  0; x;y 2 X, we show I.f /  0. Let K D Sl compactum K, we iD1 suppŒui . Since each ui is uniformly continuous on Pthe p can choose, for any " > 0, a finite decomposition K D kD1 Ek ; Ek 2 B.X/, and points k 2 Ek , 1  k  p, such that supx2K jui .x/  e u .x/j < ", where Pp Pil u . /I .x/, 1  i  l. Then jI.f /  .e ui , S vi /j  e ui .x/ D kD1 i k Ek Pl Pl Pp iD1 " iD1 .IK , jS vi j/. On the other hand, iD1 .e ui , S vi / D kD1 .IEk , Sfk / with fk .y/ D f .k ; y/. The last sum is non-negative because f  0 and Sfk  0 m-a.e. We have shown I.f /  0. By the above observation, we conclude that (1.4.6) defines a positive linear P e 0 .X  X/ D ¹f .x; y/ D l ui .x/vi .y/ W ui ; vi 2 functional on the space C iD1 e 0 .X  X/ does not depend on the C0 .X/; l  1º. The value I.f / for f 2 C manner of the expression of f . So I can be extended to a positive linear functional on C0 .X  X/. Proof of (b))(e) under assumption (1.1.7). Assume ˛G˛ is Markovian. Then by Lemma 1.4.1 there is a positive Radon measure ˛ on X  X such that Z u.x/v.y/˛ .dx; dy/ .1:4:7/ ˛.u; G˛ v/ D X X

for any Borel functions u; v 2 L2 .XI m/. Using this measure, the approximating form (1.3.16) can be rewritten as Z 1 .ˇ / E .u; u/ D ˇ .e u.x/  e u.y//2 ˇ .dx; dy/ 2 X X 1:4:8 Z 2 2 u.x/ .1  sˇ .x//m.dx/; u 2 L .XI m/; Cˇ e X

where s˛ .y/ D ˛ .X  dy/=m.dy/ and e u is any Borel modification of u. In view of Lemma 1.4.1, 0  s˛ .x/  1 m-a.e. .1:4:9/

28

1 Basic theory of Dirichlet forms

It is evident from the expression (1.4.8) and Lemma 1.3.4 that every normal contraction operates on E. Recall that a Dirichlet form is by definition a Markovian closed symmetric form. Thus all conditions (a)–(e) of Theorem 1.4.1 are satisfied for a Dirichlet form. We collect below some important properties of the Dirichlet form related to the property (e). Theorem 1.4.2. A Dirichlet form E on L2 .XI m/ possesses the following properties: (i) u; v 2 DŒE ) u _ v; u ^ v; u ^ 1 2 DŒE. 1 (ii) u; p p v 2 DŒE \ L .XI m/p) u  v 2 DŒE and E.u  v; u  v/  kuk1 E.v; v/ C kvk1 E.u; u/.

(iii) u 2 DŒE, un D ..n/ _ u/ ^ n ) un 2 DŒE and un ! u, n ! 1, with respect to E1 -metric. (iv) u 2 DŒE, u."/ D u  .."/ _ u/ ^ ", " > 0 ) u."/ 2 DŒE and u."/ ! u, " # 0, with respect to E1 -metric. (v) un ; u 2 DŒE, un ! u, n ! 1, with respect to E1 -metric and .t/ is a real function such that .0/ D 0; j.t/  .t 0 /j  jt  t 0 j; t; t 0 2 R1 ) .un /, .u/ 2 DŒE and .un / ! .u/, n ! 1, weakly with respect to E1 . If, in addition, .u/ D u, then the convergence is strong with respect to E1 . Proof. (i) By virtue of Theorem 1.4.1 (e), u 2 DŒE implies juj 2 DŒE and u ^ 1 2 DŒE. Then it suffices to note u _ v D .1=2/¹u C v C ju  vjº and u ^ v D .1=2/¹u C v  ju  vjº. (ii) By making use of formula (1.4.8), we can prove the following which is even stronger than (e): if u1 ; u2 2 DŒE; w 2 L2 .XI m/ satisfy je w .x/  w e.y/j  je u1 .x/  e u1 .y/j C je u2 .x/  e u2 .y/j; je w .x/j  je u1 .x/j C je u2 .x/j; x; p y 2 X, ;e u and w e , then w 2 DŒE and E.w; w/  for some Borel modifications e u 1 2 p p E.u1 ; u1 / C E.u2 ; u2 /. Assertion (ii) is now obtained by setting w D u  v, u1 D kuk1  v and u2 D kvk1  u. (iii) Since un is a normal contraction of u; E1 .un ; un / is uniformly bounded by E1 .u; u/. Moreover, E1 .un , G1 v/ D .un ; v/ ! .u; v/ D E1 .u; G1 v/, n ! 1, v 2 L2 .XI m/, by the formula (1.3.7). Since G1 .L2 / is dense in DŒE with metric E1 ; un weakly converges to u with respect to E1 W E1 .un ; w/ ! E1 .u; w/, 8w 2 DŒE. But then E1 .un u; un u/  2E1 .u; u/2E1 .un ; u/ ! 0; n ! 1. (iv) The proof is the same as above because u."/ is a normal contraction of u. (v) The proof is again similar to the above since .un / is a normal contraction of un .

1.4 Dirichlet forms and Markovian semigroups

29

As an application of Theorem 1.4.2 (iv), we state a useful lemma concerning the regularity of the Dirichlet form. Lemma 1.4.2. Assuming .1:1:7/, let E be a Dirichlet form on L2 .XI m/. (i) Suppose DŒE \ C1 .X/ is both E1 -dense in DŒE and uniformly dense in C1 .X/. Then E is regular. (ii) If E is regular, then for any u 2 C0 .X/ there exist un 2 DŒE \ C0 .X/; n D 1; 2; : : : ; such that suppŒun   ¹x 2 X W u.x/ ¤ 0º; n D 1; 2; : : : ; and un converges to u uniformly. Proof. For any u p 2 DŒE, there exists by assumption a sequence un 2 DŒE \ C .X/ such that E1 .u  un ; u  un /  1=n. Moreover, by Theorem 1.4.2 (iv), p1 E1 .un  e un ; u n  e un /  1=n if we take e un D .un /."/ for sufficiently small " > 0. Then e un 2 DŒE \ C0 .X/ is E1 -convergent to u. Next assume u 2 C0 .X/. By assumption, there exists for any " > 0 a function v 2 DŒE \ C1 .X/ such that ku  vk1 < ". Put e v D v ."/ . Then e v 2 DŒE \ C0 .X/; suppŒe v   ¹x 2 X W u.x/ ¤ 0º and ku  e v k1 < 2". This proves not only (i) but also (ii). Exercise 1.4.1. Assume (1.1.7). Show that, if E is regular, then DŒE \ C0 .X/ is a special standard core of E. E is called a Dirichlet form on L2 .XI m/ in the wide sense if E satisfies all conditions of the Dirichlet form on L2 .XI m/ except for the condition that DŒE is dense in L2 .XI m/. In view of the proof of Theorem 1.4.1, we see that the next statements hold. Theorem 1.4.3. (i) The family of all Markovian resolvents on L2 .XI m/ not necessarily strongly continuous and the family of all Dirichlet forms in the wide sense stand in one to one correspondence in the manner of Theorem 1:3:2. (ii) All statements of Theorem 1:4:2 remain valid for a Dirichlet form in the wide sense. In the remainder of this section, we examine the cases that strongly continuous Markovian semigroups and resolvents on L2 .XI m/ are determined by Markovian transition functions and resolvent kernels respectively. Given a measurable space .S; B/, a non-negative valued function .x; A/; x 2 S , A 2 B, is called a kernel on .S; B/ if .x;  / is a positive measure on B for each fixed x 2 S and if .; A/ is a B-measurable function for each fixed A 2 B. If an additional condition that .x; S / R 1; x 2 S , is imposed, then  is called a Markovian kernel. We write u.x/ D S u.y/.x; dy/ whenever the integral makes sense.

30

1 Basic theory of Dirichlet forms

A family ¹p t ; t > 0º of Markovian kernels on .S; B/ is said to be a Markovian transition function if p t ps u D p t Cs u;

t; s > 0; u 2 Bb :

.1:4:10/

A family ¹R˛ ; ˛ > 0º is said to be a Markovian resolvent kernel if ¹˛R˛ ; ˛ > 0º is a family of Markovian kernels on .S; B/ and R˛ u  Rˇ u C .˛  ˇ/R˛ Rˇ u D 0;

˛; ˇ > 0; u 2 Bb :

A kernel on .X; B.X// or on .X; B  .X// is called m-symmetric if Z Z u.x/.v/.x/m.dx/ D .u/.x/v.x/m.dx/ . 1/ X

.1:4:11/

.1:4:12/

X

for any non-negative measurable functions u and v. Assume that  is an m-symmetric Markovian kernel. Then Z Z 2 .u.x// m.dx/  u.x/2 m.dx/; 8u 2 Bb .X/ \ L2 .XI m/: .1:4:13/ X

X

In fact, by Schwarz inequality we have .u.x//2  1.x/  u2 .x/, which leads us to (1.4.13) because of the symmetry of . (1.4.13) means that  can be extended uniquely to a symmetric contractive operator K on L2 .XI m/. We call K a symmetric operator determined by the Markovian symmetric kernel . Given an m-symmetric Markovian resolvent kernel ¹R˛ ; ˛ > 0º on .X; B.X// or on .X; B  .X//, the family of symmetric operators ¹G˛ ; ˛ > 0º on L2 .XI m/ determined by ¹R˛ ; ˛ > 0º is readily seen to be a Markovian resolvent on L2 .XI m/ not necessarily strongly continuous. By Theorem 1.4.3, there corresponds a Dirichlet form E on L2 .XI m/ in the wide sense. E is characterized as a closed symmetric form in the wide sense satisfying R1 u 2 DŒE;

E1 .R1 u; v/ D .u; v/;

8v 2 DŒE

.1:4:14/

for any bounded Borel function u 2 L2 .XI m/. In the same way, any m-symmetric Markovian transition function determines a Markovian semigroup on L2 .XI m/ not necessarily strongly continuous. Let us give conditions which make them strongly continuous. Lemma 1.4.3. (i) Let ¹p t ; t > 0º be an m-symmetric Markovian transition function on .X; B.X// or on .X; B  .X// and ¹T t ; t > 0º be the associated Markovian semigroup on L2 .XI m/. If there exists a subfamily L of Bb .X/ \ L1 .XI m/ such that L is dense in L2 .XI m/ and lim p t u.x/ D u.x/ m-a.e. x 2 X; 8u 2 L; t #0

then ¹T t ; t > 0º is strongly continuous.

.1:4:15/

1.4 Dirichlet forms and Markovian semigroups

31

(ii) Let ¹R˛ ;˛ > 0º be an m-symmetric Markovian resolvent kernel on .X;B.X// or on .X; B  .X// and ¹G˛ ; ˛ > 0º be the associated Markovian resolvent on L2 .XI m/. If, for an L as in (i), lim ˛R˛ u.x/ D u.x/ m-a.e. x 2 X; 8u 2 L;

˛!1

.1:4:16/

then ¹G˛ ; ˛ > 0º is strongly continuous. Proof. We only give a proof of (i). For any u 2 L, p t u is L2 -convergent to u because Z Z Z 2 2 .p t u.x/  u.x// m.dx/  2 u.x/ m.dx/  2 u.x/p t u.x/m.dx/ ! 0; X

X

t # 0. Since T t is contractive, T t u is

X

L2 -convergent

to u for any u 2 L2 .XI m/.

Given an m-symmetric Markovian transition function ¹p t ; t > 0º satisfying the continuity condition (1.4.15), we can associate uniquely a Dirichlet form E on L2 .XI m/ according to Theorem 1.4.1 and Lemma 1.4.3. We call E the Dirichlet form determined by ¹p t ; t > 0º. In the next example, we compute the Dirichlet form determined by a translation invariant transition function on Rd . Example 1.4.1 (Symmetric convolution semigroup on Rd ). A system of probability measures ¹ t ; t > 0º on Rd is said to be a continuous symmetric convolution semigroup if  t  s D  t Cs ;  t .A/ D  t .A/; lim  t D ı t #0

t; s > 0; A 2 B.Rd /;

weakly,

(1.4.17) (1.4.18) (1.4.19)

R where  t  s .A/ denotes the convolution Rd  t .A  y/s .dy/ and ı is the Dirac measure concentrated at the origin. ¹ t ; t > 0º then defines a Markovian transition function p t .x; A/ on .Rd ; B.Rd // by p t .x; A/ D  t .A  x/

.1:4:20/

which is symmetric with respect to the Lebesgue measure in the sense of (1.4.12) and satisfies the continuity condition (1.4.15). The celebrated Lévy–Khinchin formula 12 under the present symmetry assumption reads as follows: ´  R  b  t .x/ D Rd e i.x;y/  t .dy/ D e t .x/ R .1:4:21/ .x/ D 12 .S x; x/ C Rd .1  cos.x; y//J.dy/; 12 Cf.

K. Itô [2].

32

1 Basic theory of Dirichlet forms

where S is a non-negative definite d  d symmetric matrix; Z d J is a symmetric measure on R n ¹0º such that

(1.4.22)

Rd n¹0º

jxj2 J.dx/ < 1: 1 C jxj2 (1.4.23)

Thus a continuous symmetric convolution semigroup ¹ t ; t > 0º is characterized by a pair .S; J / satisfying (1.4.22) and (1.4.23) through the formula (1.4.21). Let us now compute the Dirichlet form E on L2 .Rd / determined by the above ¹ t ; t > 0º. We want to show ´ R u.x/ b vN .x/ .x/dx E.u; v/ D Rd b ¯ ® R .1:4:24/ u.x/j2 .x/dx < 1 ; DŒE D u 2 L2 .Rd / W Rd jb where b u denotes the Fourier transform of u defined by Z d=2 e i.x;y/ u.y/dy; b u.x/ D .2 / Rd

x 2 Rd :

We note that b u is first defined by this integral for u 2  where  denotes the space of rapidly decreasing functions on Rd ; namely, infinitely differentiable functions whose derivatives of any order are bounded when multiplied by polynomials. Then the definition of b u is extended to any u 2 L2 .Rd / so that the Parseval formula Z b u.x/ b vN .x/dx .u; v/L2 .Rd / D Rd

remains valid. Since p t u.x/ D  t  u.x/ D b  t .x/b u.x/; x 2 Rd ; u 2 ; we have by the Parseval formula, Z 1 1 .t / .b u.x/  b  t .x/b u.x// b uN .x/dx E .u; u/ D .u  T t u; u/ D t t Rd Z 1  e t .x/ D jb u.x/j2 dx; t > 0; u 2 ; t Rd

b

1

which readily extends to any u 2 L2 .Rd / by choosing uk 2  converging to u in 2 d 2 d L R .R / as2k ! 1: The last integral in the above for u 2 L .R / increases to u.x/j .x/dx. 1/ as t # 0; which proves (1.4.24). Rd jb The R present Dirichlet form is regular. To see this, observe the expression p t u.x/ D Rd u.x C y/ t .dy/, from which we can readily conclude that p t makes

1.4 Dirichlet forms and Markovian semigroups

33

the space C1 .Rd /13 invariant and kp t u  uk1 ! 0, t # 0, for u 2 d 0 .R /. An analogous statement holds for the resolvent R˛ of p t : R˛ .x; A/ D RC1 ˛t p t .x; A/dt . Since R˛ .C0 / is E1 -dense in DŒE, we see that E is regular 0 e in view of Lemma 1.4.2 (i). When S is the unit matrix and J D 0, then .x/ D .1=2/jxj2 and the Dirichlet form (1.4.24) on L2 .Rd / reduces to the form . 12 D; H 1 .Rd // which already appeared in Example 1.2.3. As is easily seen, C01 .Rd / is a core of the form H 1 .Rd / D H01 .Rd /. The corresponding convolution semigroup is Gaussian:   1 jyj2 dy: .1:4:25/ exp   t .dy/ D 2t .2 t/d=2 When S vanishes, the Dirichlet form (1.4.24) can be rewritten as ´ R E.u; v/ D 12 Rd .Rd n¹0º/ .u.x C y/  u.x//.v.x C y/  v.x//J.dy/dx ¯ ® R DŒE D u 2 L2 .Rd / W Rd .Rd n¹0º/ .u.x C y/  u.x//2 J.dy/dx < 1 : .1:4:26/ This is a special case of Example 1.2.4. In fact, we have from (1.4.21) and (1.4.24) Z jb u.x/j2 .1  cos.x; y//J.dy/dx: E.u; u/ D Rd Rd

On the other hand, for each y 2 Rd ; the Fourier transform of function vy .x/ D u.xCy/u.x/ is b vy .x/ D b u.x/.e i.x;y/ 1/ and the Parseval formula proves that R 1 the right-hand side of the above identity is equal to 2 Rd Rd vy .x/2 J.dy/dx: The continuous symmetric convolution semigroup corresponding to .x/ D cjxj˛ with 0 < ˛  2; c > 0; is called a rotation invariant stable semigroup or symmetric stable semigroup of index ˛: The case where ˛ D 2; c D 12 ; is stated above already. For simplicity, we shall take c D 1: In particular, the case where 0 < ˛ < 2; c D 1; corresponds in the formula (1.4.21) to S D 0;

J.dy/ D

˛2˛1 . ˛Cd 2 / d=2 . 2˛ 2 /

jyj.d C˛/ dy:

Let E be the associated Dirichlet form on L2 .Rd /; namely, 8 R 0 is a positive integrable function with Fourier transform given by b ˛ .x/ D .2 /d=2 .1 C jxj2 / ˛2 : G

.1:4:29/

The Bessel potential space is then defined by ´ L˛;2 .Rd / D ¹G˛  f W f 2 L2 .Rd /º .u; v/L˛;2 .Rd / D .f; g/L2 .Rd / ; u D G˛  f; v D G˛  g; f; g 2 L2 .Rd /; .1:4:30/ so that ´ ® ¯ R 2 .1 C jxj2 /˛ dx < 1 L˛;2 .Rd / D u 2 L2 .Rd / W Rd ju.x/j O R NO .u; v/L˛;2 .Rd / D Rd u.x/ O v.x/.1 C jxj2 /˛ dx: ˛

˛

From the inequality .1 C jxj2 / 2  1 C jxj˛  21˛=2 .1 C jxj2 / 2 ; it follows that, for 0 < ˛  2, ´ ˛ DŒE D L 2 ;2 .Rd / .1:4:31/ 2 1˛=2 kuk2 ; u 2 DŒE: kukL ˛=2;2 .Rd /  E1 .u; u/  2 L˛=2;2 .Rd / It is known14 that G˛ .x/ D O.e C1 jxj / as jxj ! 1 for ˛ > 0; G˛ .x/ C2 jxjd C˛ Gd .x/ C3 log

1 jxj

as jxj ! 0 for 0 < ˛ < d; as

jxj ! 0;

for some positive constant Ci ; i D 1; 2; 3. In particular, G˛ 2 Lr .Rd / if 0 < ˛  d; 1r > 1  d˛ : We shall apply the Young inequality kg  f kq  kgkr kf kp ;

1  p; q; r  1;

1 1 1 D C  1; q p r

to g D G˛ with 0 < ˛  1 and p D 2: Here k  kp denotes the norm in Lp .Rd /: Suppose 0 < ˛  1: If d > 2˛; then kG˛ f kq  kG˛ kr kf k2 ; f 2 L2 .Rd /; with kG˛ kr < 1; for any q  2 with q1 > 12  d˛ and for r with 1r D q1 C 12 > 1 d˛ : 14 Cf.

D. R. Adams and L. I. Hedberg [1].

1.4 Dirichlet forms and Markovian semigroups

35

If d D 2˛; then the above holds for any q  2: If d < 2˛; then G˛ 2 L2 .Rd / and, by the Schwarz inequality, jG˛  f .x/j  kG˛ k2 kf k2 ; x 2 R1 ; which implies G˛  f 2 C1 .R1 / due to the continuity of the shift in L2 .R1 / and the denseness of C1 .R1 / in L2 .R1 /: By interpreting the above observation in terms of the Dirichlet form E of (1.4.28) for 0 < ˛  2; we can conclude as follows: If d > ˛; then the inequality kukq  Aq kukE1 ;

u 2 DŒE

.1:4:32/

˛ holds true for some constant Aq > 0 for any q  2 with q1 > 12  2d : If d D ˛; namely either d D ˛ D 2 or d D ˛ D 1; then (1.4.32) is valid for any q  2: If d < ˛; namely, d D 1; 1 < ˛  2; then

DŒE  C1 .R1 /;

kuk1  AkukE1 ;

u 2 DŒE;

.1:4:33/

holds where A is a positive constant. We shall see in Corollary 2.4.2 of §2.4 that ˛ (1.4.32) is also true for q with q1 D 12  2d when d > ˛: Exercise 1.4.2 (Compound Poisson process). Let c be a positive constant and  be a symmetric probability measure on Rd n ¹0º: Let ¹N t I t  0º be a Poisson process with parameter c, namely, N0 D 0; N t is right continuous and increasn ing with jump 1 and distributed as P .N t D n/ D .ctnŠ/ e ct ; n D 0; 1; 2; : : : : Let Y1 ; Y2 ; : : : ; Yn ; : : : be independent Rd -valued random variables with common distribution , which are also independent of ¹N t I t  0º: Put S0 D 0;

Sn D Y1 C Y2 C    C Yn ; n  1;

Denote by  t the distribution of X t : Show that Z t .x/ b  t .x/ D e with .x/ D c

Rd

and

X t D SN t ; t  0:

.1  cos.x; y//.dy/:

The process X t in Exercise 1.4.2 is called a symmetric compound Poisson process starting at 0: The compound Poisson process starting at x 2 Rd is defined by x C X t ; whose distribution is given by the Markovian transition function (1.4.20). Example 1.4.2 (Symmetric convolution semigroup on the unit circle). Let ƒ be the set of all real sequences  D ¹n º1 nD1 satisfying the following conditions: 0 D 0; n D n ; X .n C m  nm / n m  0 for any real sequence ¹ n º with finite support.

(1.4.34) (1.4.35)

36

1 Basic theory of Dirichlet forms

If ¹n º 2 ƒ, then n  0; 8n: n D jnj˛ ; 0 < ˛  2; n D log.jnj C 1/ are elements of ƒ. If ¹n º satisfies (1.4.34) and if n is concave and increasing for n > 0, then ¹n º 2 ƒ. The notion of a continuous symmetric convolution semigroup ¹ t º on the unit circle T D Œ0; 2 / can be defined analogously to the preceding example. Let us show that the family of all such semigroups ¹ t º t >0 and the family ƒ are in one to one correspondence by the relation b  t .n/ D e t n ;

n D 0; ˙1; ˙2; : : : ;

.1:4:36/

R 2 where b  t .n/ D 0 e inx  t .dx/. On account of the Herglotz theorem15 it suffices to prove the next assertion in order to establish the relation (1.4.36): consider a sequence ¹n º satisfying (1.4.34), then ¹n º 2 ƒ if and only if ¹exp.t n /º1 nD1 is non-negative definite for every t > 0, namely, X .1:4:37/ e t nm n m  0 for any ¹ n º with finite support. If ¹n º 2 ƒ, then X expŒt.n C m  nm / n m  0 for any ¹ n º, from which (1.4.37) follows easily. Conversely, if ¹n º satisfies (1.4.37) for all t > 0, then X X

n D 0; .1:4:38/ nm n m  0 if ¹ n º is of finite support and because the sequence n .t/ D .1=t/.1  e t n / possesses thePproperty (1.4.38) and lim t #0 n .t/ D n . Given any sequence ¹ n º with 0 D  n¤0 n , (1.4.38) reads X X X

0 n n C 0 m m C nm n m  0 n¤0

and so

m¤0

X

n;m¤0

.n C m  nm / n m  0;

n;m¤0

which implies (1.4.35). Let E be the Dirichlet form on L2 .T / determined by a continuous symmetric convolution semigroup ¹ t º t >0 on T . Then ´ P E.u; v/ D 2 1 u./ b vN ./ D1 b ® ¯ .1:4:39/ P DŒE D u 2 L2 .T / W 1 u./j2  < 1 ; D1 jb 15 Cf.

F. Riesz and B. Sz. Nagy [1; Chap. III].

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

37

u./ D 1=2  ¹ º is a sequence corresponding to ¹ t º t >0 by (1.4.36) and b Rwhere 2 i x e u.x/dx. Indeed, using Parseval’s formula, we have 0 ± X 2 ° X b  t ./jb u./j2 .1=t/.u  T t u; u/ D jb u./j2  t X X D 2 ..1  e t /=t/jb u./j2 " 2 jb u./j2  ; t # 0; u 2 L2 .T /: The Dirichlet form (1.4.39) with  D jj=2 already appeared in Example 1.2.3.

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

We have defined a Dirichlet form E as a Markovian closed symmetric form on L2 .XI m/. It is sometimes convenient to denote DŒE by F and call .E; F / a Dirichlet space relative to L2 .XI m/. F is a Hilbert space with inner product E˛ for each ˛ > 0, but F is not even a pre-Hilbert space with respect to the 0-order form E in general. In this section we introduce the notion of the transience of the Dirichlet space .E; F / which turns out to be equivalent to the transience of the corresponding Markovian semigroup. In general, the Dirichlet space F can be enlarged to a function space Fe called the extended Dirichlet space on which E can be well extended. It will be seen that Fe is a Hilbert space with respect to the 0-order form E if and only if .E; F / is transient. Let .X; B; m/ be a -finite measure space and ¹T t ; t > 0º be a strongly continuous Markovian semigroup on L2 .XI m/. We put Z t Ts f ds; f 2 L2 .XI m/; .1:5:1/ St f D 0

the integral being defined as the Bochner integral in L2 .XI m/. Both T t and S t are bounded symmetric operators on L2 .XI R m/: kT t f kL2  kf kL2 ; kS t f kL2  1 \ L2 . Then tkf k : Let f 2 L 2 L K jT t f .x/jm.dx/  .T t IK ; jf j/  R jf .x/jm.dx/ for any compact set K. Letting K " X, kT t f kL1  kf kL1 : X Similarly, we have kS t f kL1  t kf kL1 : Therefore, T t and S t on L1 \ L2 can be uniquely extended to bounded linear operators on L1 .XI m/ and kT t f kL1  kf kL1 ;

kS t f kL1  t kf kL1 ;

f 2 L1 .XI m/:

.1:5:2/

The extended operators are positive; moreover, S t f .x/  S t 0 f .x/ m-a.e. whenever t < t 0 , f 2 L1C .XI m/. Here L1C .XI m/ D ¹f 2 L1 .XI m/ W f .x/  0 m-a.e.º. Obviously the resolvent operators G˛ ; ˛ > 0, have analogous properties.

38

1 Basic theory of Dirichlet forms

Let us put Gf .x/ D lim SN f .x/ D lim G1=N f .x/ . C1/; f 2 L1C .XI m/: N !1

N !1

.1:5:3/ The limit exists m-a.e. but may take the value C1. We call the Markovian semigroup ¹T t ; t > 0º transient if Gf .x/ < C1 m-a.e. for anyf 2 L1C .XI m/:

.1:5:4/

Lemma 1.5.1. A Markovian semigroup ¹T t ; t > 0º is transient if and only if there exists a function g 2 L1 .XI m/ such that g is strictly positive m-a.e. on X and Gg.x/ < 1 m-a.e. To prove this, we need another lemma. Lemma 1.5.2 (Hopf’s maximal ergodic inequality).R For f 2 L1 .XI m/ and h > 0, let Eh D ¹x 2 X W supn Snh f .x/ > 0º. Then Eh Sh f .x/m.dx/  0. Proof. Let ® ¯ ® ¯ Ehn D x 2 X W max Sh f .x/ > 0 D x 2 X W max .Sh f /C .x/ > 0 : 1n

1n

Here we denote u _ 0 by uC for a function u on X. Then, for x 2 Ehn ; we have Sh f .x/ C max .S.C1/h f  Sh f /C .x/  max .Sh f /C .x/: 1n

1n

In view of the positivity of T t , max .S.C1/h f  Sh f /C .x/  Th Πmax .Sh f /C .x/:

1n

1n

Hence, Z Ehn



Sh f .x/m.dx/ Z

®

Ehn

max .Sh f /C .x/  Th

1n



¯ max .Sh f /C .x/ m.dx/

1n







 max .Sh f /C L1  Th max .Sh f /C L1  0: 1n

1n

Letting n tend to infinity, we have the desired inequality.

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

39

Proof of Lemma 1:5:1. Let g be a function satisfying the condition in Lemma 1.5.1. For any function f 2 L1C .XI m/ and for any a > 0 andRh > 0, we put A D ¹x 2 X W supn Snh .f ag/.x/ > 0º. Then, by Lemma 1.5.2, A Sh .f ag/d m  0. Since the set B DR ¹x 2 X W RGf .x/ D 1º Ris contained in Rthe set A up to an m-negligible set, h X f dRm  A Sh f d m  a A Sh gd m  a B Sh gd m. Hence R .1=a/ X f d m  .1= h/ B\K Sh .g ^ N /d m for any set K 2 B with m.K/ < 1 2 and integer N . Since R .1= h/Sh .g ^ N / is L -convergent to g ^ N as h # 0, R .1=a/ X f d m R B\K .g ^ N /d m: After letting K " X, N " 1 and then a " 1, we get B gd m D 0, which implies m.B/ D 0: Let ¹T t ; t > 0º be a strongly continuous Markovian semigroup on L2 .XI m/ and .E; F / be the associated Dirichlet space relative to L2 .XI m/. Lemma 1.5.3. For any non-negative g 2 L1 .XI m/ \ L2 .XI m/, sZ .juj; g/ g  Ggd m . C1/: D sup p E.u; u/ X u2F Proof. First we notice that, for any g 2 L2 .XI m/; S t g 2 F and .1:5:5/ E.S t g; u/ D .g  T t g; u/; 8u 2 F : R t Cs Rs Indeed, S t g  Ts S t g D  t Tv gdv C 0 Tv gdv and 1s .S t g  Ts S t g; S t g/ converges as s # 0 to the finite limit .g; S t g/  .T t g; S t g/ proving that S t g 2 F . The same calculation p gives (1.5.5). If supu2F .juj; g/= E.u; u/ D c < 1, then p p .S t g; g/  c E.S t g; S t g/  c .S t g; g/ qR p and so .S t g; g/  c. Letting t " 1, we get X gGgd m  c: R R1 R Conversely, suppose that X gGgd m < 1. Since X gGgd m D 0 .Ts g; g/ds and .Ts g; g/ D .Ts=2 g; Ts=2 g/ is non-increasing as s " 1, we must have lims"1 .Ts g; g/ D 0. On account of (1.5.5), we then have .juj; g/ D E.juj; S t g/ C .juj; T t g/ p p p p  E.S t g; S t g/ E.u; u/ C .T t g; T t g/ .u; u/ p p p p  .S t g; g/ E.u; u/ C .T2t g; g/ .u; u/ sZ p gGgd m E.u; u/; t " 1; ! X

for any u 2 F .

40

1 Basic theory of Dirichlet forms

Any strongly continuous transient Markovian semigroup ¹T R t ; t > 0º admits a strictly positive bounded m-integrable function g such that X g  Ggd m  1. We call such g a reference function of the transient semigroup ¹T t ; t > 0º. To construct g, it suffices to take a strictly positive bounded measurable function f 2 R L1 .XI m/ with X f d m D 1 and let g D Rf =.Gf _ 1/: g isRdominated by f Then we have X g  Ggd m  X f  Ggd m  R Rand strictly positive m-a.e. Gf  .f =Gf /d m D f d m D 1. X X We say that a Dirichlet space .E; F / relative to L2 .XI m/ is transient if there exists a bounded m-integrable function g strictly positive m-a.e. on X such that Z p jujgd m  E.u; u/; 8u 2 F : .1:5:6/ X

The function g above is called a reference function of the transient Dirichlet space .E; F /. As an immediate consequence of Lemma 1.5.3, we have Theorem 1.5.1. Let ¹T t ; t > 0º be a strongly continuous Markovian semigroup on L2 .XI m/ and .E; F / be the associated Dirichlet space relative to L2 .XI m/. Then ¹T t ; t > 0º is transient if and only if .E; F / is transient. In this case, both have common reference functions. Given a Dirichlet space .E; F / relative to L2 .XI m/, we denote by Fe the family of m-measurable functions u on X such that juj < 1 m-a.e. and there exists an E-Cauchy sequence ¹un º of functions in F such that limn!1 un D u m-a.e. We call ¹un º as above an approximating sequence for u 2 Fe . Fe is a linear space containing F . In the rest of this section, we assume (1.1.7). Theorem 1.5.2. Let .E; F / be a Dirichlet space relative to L2 .XI m/. (i) For any u 2 Fe and its approximating sequence ¹un º, the limit E.u; u/  limn!1 E.un ; un / exists and does not depend on the choice of the approximating sequence for u. (ii) Extend the expression .1:4:8/ of the approximating form associated with .E; F / to any Borel function u on X: Z Z Z ˇ E .ˇ / .u; u/ D .u.x/u.y//2 ˇ .dxdy/Cˇ u.x/2 .1sˇ .x//m.dx/: 2 X X X u of u, E ˇ .e u;e u / increases to If u 2 Fe then, for any Borel modification e E.u; u/ as ˇ " 1. (iii) F D Fe \ L2 .XI m/.

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

41

Proof. Let u 2 Fe and ¹un º be its approximating sequence. The limit in (i) obviously exists by the triangle inequality. u satisfy limn!1 e un .x/ We may assume that some Borel modifications e un and e De u.x/ for any x 2 X: Then by Fatou’s lemma un  e u;e un  e u /  lim inf E .ˇ / .e un  e um ;e un  e um / E .ˇ / .e m!1

 lim E.un  um ; un  um / m!1

which can be made arbitrarily small for sufficiently large n. In particular un ;e un / D E .ˇ / .e u;e u /. Hence E .ˇ / .e u;e u / is non-decreasing with ˇ. limn!1 E .ˇ / .e Furthermore, the inequality jE.u; u/1=2  E .ˇ / .e u;e u /1=2 j un ;e un /1=2 j  jE.u; u/1=2  E.un ; un /1=2 j C jE.un ; un /1=2  E .ˇ / .e un  e u;e un  e u /1=2 C E .ˇ / .e implies that limˇ !1 E .ˇ / .e u;e u / D E.u; u/: u;e u / D 0 and We have shown (ii). In particular, if u D 0 m-a.e., then E .ˇ / .e E.u; u/ D 0. This means the second statement in (i). (iii) is an immediate consequence of (ii). By virtue of Theorem 1.5.2, E can be well extended to Fe as a non-negative definite symmetric bilinear form. We call .Fe ; E/ the extended Dirichlet space of .E; F /. As was seen by Lemma 1.3.3, the Markovian semigroup ¹T t ; t > 0º on L2 .XI m/ has the properties T t .F /  F ;

E.T t u; T t u/  E.u; u/;

8u 2 F

lim kT t u  ukE D 0; t #0

where kvkE D space .Fe ; E/.

p

E.v; v/; v 2 F . This can be extended to the extended Dirichlet

Lemma 1.5.4. T t can be uniquely extended to a linear operator on .Fe ; E/ such that u  T t u 2 F for u 2 Fe and E.T t u; T t u/  E.u; u/;

u 2 Fe ;

lim kT t u  ukE D 0: t #0

.1:5:7/

42

1 Basic theory of Dirichlet forms

Proof. For u 2 Fe , take an approximating sequence ¹un º for it. (1.5.7) applied to ¹un º yields that ¹T t un º is E-Cauchy. Since .T t w; T t w/  .w; T t w/ for any w 2 L2 , we get, by letting un;m D un  um ; 1 2 k.un  um /  T t .un  um /kL 2 .m/ t 1 1 D .un;m  T t un;m ; un;m /  .un;m  T t un;m ; T t un;m / t t 1  ..un;m  T t un;m ; un;m /  E.un  um ; un  um /: t Consequently, ¹un  T t un º is Cauchy in L2 .XI m/. In particular, a subsequence ¹T t unk º converges a.e. to a function v, which can be easily seen to be independent of the choice of the approximating sequence ¹un º. We let T t u D v. Then T t u 2 Fe and T t unk is its approximating sequence and u  T t u 2 Fe \ L2 .XI m/ D F . Clearly (1.5.7) is extended to u 2 Fe . The last assertion follows from kT t u  ukE  kT t un  un kE C kT t .u  un /kE C ku  un kE  kT t un  un kE C 2ku  un kE ; if we let first t ! 0 and then n ! 1. From now on we study some basic properties of the extended Dirichlet space in transient case. Lemma 1.5.5. If a Dirichlet space .E; F / relative to L2 .XI m/ is transient, then its extended Dirichlet space Fe is complete with metric E. Proof. Observe that, for u 2 Fe and its approximating sequence ¹un º, E.u  un ; u  un / D liml!1 E.ul  un ; ul  un / and consequently un is E-convergent to u as n ! 1. Suppose that .E; F / is transient. Let ¹un º be an E-Cauchy sequence of functions of Fe . By the above observation, there exists an E-Cauchy sequence ¹vn º  F such that limn!1 E.un  vn ; un  vn / D 0: According to the assumption of transience of .E; F /, the inequality (1.5.6) applied to u D vn  vm implies that the sequence ¹vn º forms a Cauchy sequence in L1 .XI gd m/. Hence there exists a subsequence ¹vnl º of ¹vn º which converges m-a.e. to a function u. Then u 2 Fe and E.un  u; un  u/1=2  E.un  unl ; un  unl /1=2 C E.unl  vnl ; unl  vnl /1=2 C E.vnl  u; vnl  u/1=2 : By letting l ! 1 and then n ! 1, we see that un is E-convergent to u as n ! 1.

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

43

Theorem 1.5.3. A pair .Fe ; E/ is the extended Dirichlet space of a transient Dirichlet space relative to L2 .XI m/ if and only if the following conditions are satisfied: .˛/ Fe is a real Hilbert space with inner product E, .ˇ/ there exists an m-integrable bounded function g strictly positive m-a.e. such that Fe  L1 .XI g  m/ and Z p ju.x/jg.x/m.dx/  E.u; u/; 8u 2 Fe ; .1:5:8/ X

. / Fe \ L2 .XI m/ is dense both in L2 .XI m/ and in .Fe ; E/, .ı/ every normal contraction operates on .Fe ; E/; i.e., u 2 Fe ; v is a normal contraction of u ) v 2 Fe ; E.v; v/  E.u; u/. In this case, .Fe \ L2 .XI m/; E/ is a Dirichlet space relative to L2 .XI m/ and .Fe ; E/ is its extended Dirichlet space. Proof. As for the proof of the “only if” part, properties .˛/; .ˇ/ and . / are clear from Theorem 1.5.1, Theorem 1.5.2 and Lemma 1.5.5. To show property .ı/, suppose that v is a normal contraction of u 2 Fe . Take an approximating sequence un for u and set wn D .v ^ jun j/C . Then it is easy to see that E .ˇ / .wn ; wn /  E .ˇ / .e v ;e v / C E .ˇ / .un ; un /  E.u; u/ C E.un ; un /: Since wn 2 L2 .XI m/, wn belongs to F and E.wn ; wn / is uniformly bounded in n. Therefore the Cesàro mean of a subsequence of wn is E-convergent to some w 2 Fe .16 But v C D limn!1 wn D w 2 Fe and so v D v C  v  2 Fe : The inequality E.v; v/  E.u; u/ is obvious from Theorem 1.5.2 (ii). It is easy to prove the “if” part, namely, the final statement in this theorem. On account of Theorem 1.5.3, we may call a pair .Fe ; E/ satisfying .˛/; .ˇ/; . / and .ı/ a transient extended Dirichlet space with reference measure m. The function Gf has been defined for f 2 L1C .XI m/ by (1.5.3). We extend the potential operator G for any non-negative m-measurable function f on X by Gg.x/ D lim G.f ^ .n//.x/ for m-a.e. x 2 X; n!1

where  is a fixed strictly positive bounded m-integrable function on X: Note that, for f 2 L1C .XI m/; the right-hand side in the above coincides with Gf .x/ of (1.5.3) m-a.e. due to the exchangeability of monotone limits. 16 Cf.

F. Riesz and B. Sz. Nagy [1; §38].

44

1 Basic theory of Dirichlet forms

Theorem 1.5.4. (i) The family of all transient extended Dirichlet spaces .Fe ; E/ with reference measure m and the family of all strongly continuous transient Markovian semigroups ¹T t ; t > 0º on L2 .XI m/ are in one to one correspondence. The correspondence is characterized by Rthe following relation; for any non-negative measurable function f such that X f  Gf d m < 1, f  v 2 L1 .XI m/ 8v 2 Fe ;

8v 2 Fe : .1:5:9/ In this case, the function Gf does not depend on the choice of  in its definition above up to the m-equivalence. Gf 2 Fe ;

E.Gf; v/ D .f; v/;

(ii) For a reference function g of R a transient Dirichlet form .E; F /; let L D ¹f D h  g W h 2 bB.X/º: Then X f  Gf d m < 1 for any f 2 LC and G.L/ is dense linear subspace of the extended Dirichlet space .Fe ; E/: Proof. Let a transient semigroup ¹T t ; t > 0º and an extended Dirichlet space .Fe ; E/ correspond to each other as in Theorem 1.5.1 R and Theorem 1.5.3. First consider a non-negative f 2 L2 .XI m/ such that X f Gf d m < 1: By (1.5.5) we have for t > t 0 E.S t f  S t 0 f; S t f  S t 0 f /  .S t f; f /  .S t 0 f; f / C .S t 0 f; T t f  T t 0 f / R which converges as t; t 0 ! 1 to zero because .S t f; f / ! X f Gf d m and R t Ct 0 .S t 0 f; T t f / D t .Tu f; f /du ! 0. Hence ¹Sn f º is an E-Cauchy sequence which converges to Gf m-a.e. and hence Gf 2 Fe . Now (1.5.5) again leads us to the equation in (1.5.9). Next take v 2 Fe with its approximating sequence ¹vn º  F : Since Z jv`  vn jf d m D E.Gf; jv`  vn j/  kGf kE kv`  vn kE ; we see that ¹vn º is L1 .XI f  m/-Cauchy and converges to v m-a.e. Therefore we have the second property in (1.5.9). By letting n ! 1 in the equation of (1.5.9) for v D vn ; we get the same equation for v 2 Fe : R Now consider a non-negative measurable function f such that X f Gf d m < 1. Put fn D f ^ .n/.2 L2 .XI m//. Then, by what has just been proved, we have Z Z fn Gfn d m  fk Gfk d m; E.Gfn  Gfk ; Gfn  Gfk /  X

X

n > k, which converges to zero as n; k ! 1. Therefore we are led to (1.5.9) for f from that for fn .

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

45

The equation in (1.5.9) in particular implies that the function Gf 2 Fe depends only on f and does not depend on a particular choice of  in its definition. The first assertion of (ii) is immediate from Lemma 1.5.3. The second assertion of (ii) is also clear because any f 2 L satisfies (1.5.9). This particularly implies that, for a given transient semigroup ¹T t ; t > 0º, a transient extended Dirichlet space .Fe ; E/ satisfying the relation in (i) is unique. Conversely, given a transient extended Dirichlet space .Fe ; E/, the transient semigroup satisfying the relation in (i) is unique. First observe that Gf D limn!1 G1=n f m-a.e. for a non-negative function f 2 L1 \ L2 and so the resolvent equation extends to Gf D G˛ f C ˛GG˛ f . Assume further that R f Gf d m < 1. Then X Z Z Z 1 1 G˛ f  GG˛ f d m  G˛ f  Gf d m D f GG˛ f d m ˛ X ˛ X X Z 1 f Gf d m < 1:  2 ˛ X In view of (1.5.9), GG˛ f 2 Fe and E.GG˛ f; v/ D .G˛ f; v/; 8v 2 Fe . Hence G˛ f 2 Fe , E.G˛ f; v/ D E.Gf; v/  ˛E.GG˛ f; v/ D .f; v/  ˛.G˛ f; v/;

8v 2 Fe ;

namely, G˛ f 2 F D Fe \ L2 .XI m/; E˛ .G˛ f; v/ D .f; v/ 8v 2 F . The last relation holds for any functions in the space L appearing in (ii). Since L is dense in L2 .XI m/, ¹G˛ ; ˛ > 0º is uniquely determined by .Fe ; E/ and so is ¹T t ; t > 0º. As a consequence of the preceding two theorems, we can readily extend Theorem 1.4.2 to a transient extended Dirichlet space. Corollary 1.5.1. Let .Fe ; E/ be a transient extended Dirichlet space with reference measure m. Then the statements (i), (iii), (iv) and (v) of Theorem 1:4:2 are valid with DŒE and E1 being replaced by Fe and E respectively. Further the product of bounded functions in Fe is again in Fe . To verify the last assertion, consider bounded u; v 2 Fe and choose un ; vn 2 F which are E-convergent to u; v respectively. Due to the present version of Theorem 1.4.2 (v), we may suppose that un ; vn are uniformly bounded. Then E.un  vn ; un  vn / are uniformly bounded in view of Theorem 1.4.2 (ii). The Cesàro means of a subsequence of ¹un  vn º are E-Cauchy and convergent to u  v m-a.e. Hence u  v 2 Fe . Here we present a criterion for a function space to become the extended Dirichlet space of a given transient Markovian semigroup.

46

1 Basic theory of Dirichlet forms

Theorem 1.5.5. Let ¹T t ; t > 0º be a strongly continuous transient Markovian semigroup on L2 .XI m/ and .G ; a/ be a real Hilbert space satisfying the following conditions: (i) G is a collection of m-measurable functions on X, two functions being identified if they coincide m-a.e. (ii) There exists a linear subspace L of L1 .XI m/ such that (a) f 2 L H) jf j 2 L; 1 (b) If R an m-measurable function v satisfies f  v 2 L .XI m/ and X f vd m D 0 for any f 2 LC ; then v D 0 m-a.e.

(iii) For any f 2 LC and any v 2 G ; Z Gf 2 G ;

a.Gf; v/ D

f vd m:

.1:5:10/

X

Then .G ; a/ D .Fe ; E/ the extended Dirichlet space of ¹T t ; t > 0º. Proof. Put G.L/ D ¹Gf1  Gf2 W f1 ; f2 2 Lº: Then, by (ii) (b) and (iii), G.L/ is a dense linear subspace of .G ; a/ and Z f  Gf d m D a.Gf; Gf / < 1 for every f 2 LC : X

Hence, by virtue of Theorem 1.5.4, for every f 2 LC and v 2 Fe ; Z f vd m: Gf 2 Fe ; f  v 2 L1 .XI m/ and E.Gf; v/ D X

By the assumption (ii), we then see that G.L/ is dense in .Fe ; E/: Since a D E on G.L/; we get the desired conclusion. A typical example of the space L satisfying the conditions in (ii) of Theorem 1.5.5 is given by L D ¹f D h  g W h 2 bB.X/º for a reference function g of the semigroup ¹T t I t > 0º: Theorem 1.5.5 will be utilized in Example 1.5.2 and §6.2. A. Beurling and J. Deny17 first introduced the notion of Dirichlet space. In the present context, Beurling–Deny’s original notion of a (regular) Dirichlet space is nothing but a transient extended Dirichlet space .Fe ; E/ which satisfies instead of .ˇ/ and . / the stronger conditions .ˇ/0 and . /0 described below under the topological assumption (1.1.7) on X and m: 17 A.

Beurling and J. Deny [2].

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

47

.ˇ/0 Fe  L1loc .XI m/ and for any compact set K  X Z p ju.x/jm.dx/  CK E.u; u/; 8u 2 Fe ; K

CK being a positive constant. . /0 .Fe ; E/ is regular; i.e., Fe \ C0 .X/ is dense both in .Fe ; E/ and in C0 .X/. Notice that the condition .ˇ/0 is the same as saying that the transient Dirichlet space .E; F / possesses as its reference function g a strictly positive function of C1 .X/ \ L1 .XI m/. The next theorem is an immediate consequence of Lemma 1.5.3 and Theorem 1.5.4 (ii). Theorem 1.5.6. Assume .1:1:7/ for X and m. Let a transient extended Dirichlet space .Fe ; E/ with reference measure m and a strongly continuous transient Markovian semigroup ¹T t ; t > 0º correspond to each other as in Theorem 1:5:4. Then a necessary and sufficient condition that .E; F / possesses the stronger property .ˇ/0 is that   Z 1 Z .T t f; f /dt < 1; 8f 2 C0C .X/: f  Gf d m D 0

Example 1.5.1. Consider Example 1.2.3 and assume that the Euclidean domain D is bounded. Then the Poincaré inequality .u; u/ 

l2 D.u; u/; 2d

u 2 H01 .D/

.1:5:11/

holds with l being the diameter of D. In fact, for u 2 C01 .D/, this follows immediately from the bound  Z 1 @u 2 2 0 dxi ; 1  i  d; u.x/  .xi  xi / 1 @xi where xi0 is the infimum of the support of u as a function of xi . Accordingly the Dirichlet form . 12 D; H01 .D// on L2 .D/ is transient. Since H01 .D/ is already complete with respect to 12 D, H01 .D/ is the same as its extended Dirichlet space. The operator G is bounded on L2 .D/. Let us examine the operator G in the simplest case that D D B the ball with center 0 and radius > 0. Denote by … .x; d / the harmonic measure … .x; d / D

2  jxj2  .d /; jx  jd

x 2 B ;

48

1 Basic theory of Dirichlet forms

where  is the uniform probability measure on the boundary @B . The Green function for B is then defined by Z . / … .x; d /v.  y/; x; y 2 B ; R .x; y/ D v.x  y/  @B

where v.x/ D

8 1 1 ˆ <  log jxj d ˆ : . 2 1/

2 d=2

if d D 2 1 jxjd 2

if d  3:

We show that the operator G for the space . 12 D; H01 .B // is given by the kernel . / y/. For any bounded smooth function f on B , the function R. / f .x/ D RR .x; . / .x; y/f .y/dy is easily seen to satisfy the Poisson equation B R 1  R. / f D f 2

on B ;

R. / f D 0 on @B

and according to the divergence theorem R. / f 2 H 1 .B /;

1 D.R. / f; g/ D .f; g/B ; 2

8g 2 C01 .B /:

On account of the criterion .G :3/ stated after (1.2.14), the zero extension u of R. / f to Rd is an element of H 1 .Rd /. Further u .x/ D u.x/ is D-convergent to u as  # 1. Since u for  > 1 vanishes on a neighbourhood of @B , it can be D-approximated by C01 .B /-functions using a mollifier as in Exercise 1.2.1. Hence R. / f 2 H01 .B / as was to be checked. Example 1.5.2. Consider a symmetric convolution semigroup ¹ t ; t > 0º on Rd as in Example 1.4.1. The associated semigroup ¹T t ; t > 0º and Dirichlet space .E; R 1 F / are defined by (1.4.20) and (1.4.24) respectively. We put .A/ D 0  t .A/dt . C1/ and assert that the following four conditions are equivalent: ¹T t ; t > 0º is transient. .1:5:12/ .A/ < 1 for any compact set A  Rd : Z 1 .T t f; f /dt < 1; 8f 2 C0C .Rd /:

.1:5:13/ .1:5:14/

0

1= ./ is locally integrable on Rd : .1:5:15/ R1 Assume (1.5.13). Then 0 p t .x; A/dt D .Ax/ is finite for any compact A and for almost all x. This means (1.5.14), which in turn implies (1.5.12). Obviously,

1.5

Transience of Dirichlet spaces and extended Dirichlet spaces

49

(1.5.13) follow from (1.5.12). Let A be any compact set. Choose f 2 C0C .Rd / b.x/j  1; 8x 2 A; e.g. f .x/ D  2dd Qd 1.a;a/ .xi /; x 2 Rd ; such that jf iD1 .2a/ for a sufficiently small a > 0: Then Z Z Z 1 Z 1 b./j2 d jf N  d D .T t f ; fO/dt D .T t f; f /dt; ./ ./ 0 0 A Rd

b

which R is finite under the condition (1.5.14). On the other hand, for g./ D C Rd f . C /f ./d, with C D .2 /d=2 , (1.5.15) implies that ³ ³ Z T ²Z Z T ²Z b.x/j2  t .dx/ dt D lim .A/  lim jf b g .x/ t .dx/ dt T !1 0

Z

Rd

T

³

²Z

D C lim

T !1 0

Rd

T !1 0

g./b  t ./d  dt D C

Z

Rd

Rd

g./ d  < 1: ./

In view of Theorem 1.5.6 and (1.5.14), the extended Dirichlet space .Fe ; E/ associated with a transient convolution semigroup always satisfies the Beurling– Deny condition .ˇ/0 . It satisfies the regularity condition . /0 as well because .E; F / is regular as we saw in Example 1.4.1. Furthermore, the following explicit expression is known: 8 0º: We then have G .D/ D linear span of He1 .D/ and N ;

.1:5:25/

He1 .D/ does not contain non-zero constant functions on D, just as in the whole space case. To show (1.5.25), take any u 2 G .D/. In view of the property (G :3) stated in Example 1.2.3, we may suppose that there is a subset N of the boundary @D D ¹.x 0 ; xd / W xd D 0º with zero .d  1/-dimensional Lebesgue measure such that, for each .x 0 ; 0/ 2 @DnN , u.x 0 ; xd / is as a function of xd absolutely continuous on .0; 1/ with a Radon–Nikodym derivative, say g.x 0 ; xd /, being square integrable on .0; 1/ with respect to xd . If we set ´ R1 u.x 0 ; 1/  0 g.x 0 ; xd /dxd ; .x 0 ; 0/ 2 @D n N 0 u.x ; 0/ D 0; .x 0 ; 0/ 2 N; then u is the boundary function of u: u.x 0 ; 0/ D lim u.x 0 ; xd /; xd #0

8x 2 @D n N:

u.x/ to be u.x/ We now extend u to a function b u on Rd by reflection; we define b u satisfies the if xd > 0, u.x/ if xd D 0 and u.x 0 ; xd / if xd < 0. Clearly b d d conditions of .G :3/ on R , b u 2 G .R / and D.b u;b u/ D 2D.u; u/. 1 to This extension enables us to reduce the properties of G .D/ and  1 H .D/ d 1 d those of G .R / and H .R /. For instance, the Dirichlet form 2 D; H 1 .D/

1.6

Global properties of Markovian semigroups

53

is transient because inequality (1.5.6) or stronger one (1.5.20) holding on Rd is inherited to the half space D. Furthermore, for any non-constant u 2 G .D/, b u can be D-approximated on Rd by some functions vn 2 H 1 .Rd / by virtue of (1.5.24). Hence u is D-approximated on D by vn jD 2 H 1 .D/ and consequently u 2 He1 .D/. This along with the transience of H 1 .D/ implies the desired relation (1.5.25). In the same way, we see that C01 .D/, the space of restrictions to D offunctions  in C01 .Rd /, is dense in H 1 .D/. This proves that the form 12 D; H 1 .D/ regarded as a Dirichlet form on L2 .D/ rather than on L2 .D/ is regular. In the present half space case, we further get from (1.5.23) and (1.5.25) the orthogonal decomposition 1 .D/ ˚ HV .D/ He1 .D/ D H0;e

.1:5:26/

where HV .D/ denotes the space of all harmonic functions on D belonging to the space He1 .D/: The two-dimensional cases require a different approach and will be studied in Example 1.6.2.

1.6

Global properties of Markovian semigroups

In the preceding section, the transience of Markovian semigroups has been studied in terms of Dirichlet spaces and their extended ones. In this section, we explore in the same spirit more about global properties of not only transience but also recurrence, irreducibility and conservativeness. We continue to work with a -finite measure space .X; B; m/ and a strongly continuous Markovian semigroup ¹T t ; t > 0º on L2 .XI m/ and keep those relevant notions introduced so far. In particular .E; F / denotes the Dirichlet space relative to L2 .XI m/ associated with ¹T t ; t > 0º and .Fe ; E/ denotes the extended Dirichlet space of .E; F /. An m-measurable set A  X is said to be (T t -)invariant if T t .1A f / D 1A T t f m-a.e. for any f 2 L2 .XI m/ and t > 0. Lemma 1.6.1. The following conditions are equivalent to each other for an mmeasurable set A: (i) A is invariant. (ii) T t .1A f / D 0 m-a.e. on X n A for any f 2 L2 .XI m/ and t > 0. (iii) G˛ .1A f / D 0 m-a.e. on X n A for any f 2 L2 .XI m/ and ˛ > 0. The family of invariant sets is closed under the operation of taking complement, countable union and countable intersection.

54

1 Basic theory of Dirichlet forms

Proof. If (ii) is satisfied by A, then X n A has the same property owing to the symmetry of T t . Hence T t .1A f / D 1A T t .1A f / D 1A .T t f  T t .1X nA f // D 1A T t f proving (i). The rest of the proof is clear. Theorem 1.6.1. The following conditions are equivalent to each other for an mmeasurable set A: (i) A is invariant. (ii) 1A  u 2 F for any u 2 F and E.u; v/ D E.1A u; 1A v/ C E.1X nA u; 1X nA v/;

u; v 2 F :

.1:6:1/

(iii) 1A  u 2 Fe for any u 2 Fe and the equality (1.6.1) holds for any u; v 2 Fe . Proof. If A is invariant, then .u; .I  T t /u/ D .1A u; .I  T t /1A u/ C .1X nA u; .I  T t /1X nA u/ and we see by Lemma 1.3.4 that (ii) is valid. Suppose conversely that (ii) is satisfied. Then, by setting 1A u in place of u in (1.6.1), we see E.1A u; v/ D E.1A u; 1A v/ D E.u; 1A v/, u; v 2 F . For f 2 L2 .XI m/, we have E˛ .G˛ 1A f; v/ D .1A f; v/ D .f; 1A v/ D E˛ .G˛ f; 1A v/ D E˛ .1A G˛ f; v/;

v 2F;

and accordingly G˛ 1A f D 1A G˛ f the property (i). (iii) is evidently equivalent to (ii). Lemma 1.6.2. For any g 2 L1C .XI m/, the sets ¹x W Gg.x/ D 1º and ¹x W Gg.x/ D 0º are invariant. These sets are m-a.e. independent of the choice of the strictly positive g 2 L1C .XI m/. Proof. For g 2 L1C let Bn D ¹x W Gg.x/  nº and B D ¹x W Gg.x/ < 1º. Then, for any f 2 L1C , the inequality .T t .1Bn f /; Gg/ D .1Bn f; T t Gg/  .1Bn f; Gg/  n.f; 1/ implies that T t .1Bn f / D 0 on X n B. Letting n " 1, we get T t .1B f / D 0 m-a.e. on X n B. Similarly the set ¹xI Gg.x/ D 0º is an invariant set.

1.6

Global properties of Markovian semigroups

55

Suppose that gi .x/; i D 1; 2, are strictly positive functions of L1C .XI m/. Let Bi D ¹xI Ggi .x/ < 1º. Then B1 [ B2 is an invariant set on which G.g1 ^ g2 / is finite. Hence, by virtue of Lemma 1.5.1, the restriction of T t to B1 [ B2 defines a transient symmetric Markovian semigroup with respect to the restriction of m to B1 [ B2 . But then Ggi < 1 m-a.e. on B1 [ B2 for i D 1; 2 and hence B1 D B2 m-a.e. For a fixed strictly positive function g 2 L1 .XI m/, we call the invariant sets Xd  ¹x W Gg.x/ < 1º and Xc  ¹x W Gg.x/ D 1º the dissipative part and conservative part of X relative to ¹T t ; t > 0º respectively. Lemma 1.6.3. For f 2 L1C .XI m/, let B D ¹x 2 Xc W Gf .x/ < 1º: Then f D 0 and Gf D 0 m-a.e. on B. Proof. Take a strictly positive function g 2 L1C .XI m/. Since Gg D 1 m-a.e. on Xc , the set B is contained in the set A D ¹x 2 X W supn Snh .g  af /.x/ > 0º up to anR m-negligibleRset for any h > 0 and R a > 0. By Lemma 1.5.2, we and hence have Ra1 X gd m  h1 B Sh f d m B Sh f d m D 0, from which folR lows B Gf d m D 0 and also B f d m D 0 as in the last part of the proof of Lemma 1.5.1. We call the Markovian semigroup ¹T t ; t > 0º or the Dirichlet form E recurrent if Gf .x/ D 0 or 1

for m-a.e. x 2 X for any f 2 L1C .XI m/:

.1:6:2/

The Markovian semigroup ¹T t ; t > 0º or the Dirichlet form E is called irreducible if any T t -invariant set B satisfies either m.B/ or m.X n B/ D 0. Lemma 1.6.4. (i) ¹T t ; t > 0º is transient if and only if Xd D X m-a.e. (ii) ¹T t ; t > 0º is recurrent if and only if Xc D X m-a.e. (iii) Suppose ¹T t ; t > 0º is irreducible, then it is either transient or recurrent. In the recurrent case, Gf .x/ D 1 m-a.e. on X for any non-negative measurable function f such that m.¹x W f .x/ > 0º/ > 0. Proof. (i) is evident by Lemma 1.5.1. As in the last part of the proof of Lemma 1.5.1, we see that, for any f 2 L1C .XI m/, f vanishes m-a.e. on ¹x 2 X W Gf .x/ D 0º. This proves the “only if” part of (ii) and the second statement of (iii). The “if” part of (ii) follows from Lemma 1.6.3. The first statement of (iii) is an immediate consequence of (i) and (ii).

56

1 Basic theory of Dirichlet forms 2 For any f 2 L1 C .XI m/, take a sequence ¹fn º  LC .XI m/ such that fn " f .1/

.2/

m-a.e. and let T t f D limn!1 T t fn : If ¹fn º and ¹fn º are such sequences, then .1/

lim T t fk

k!1

.1/

^ fn.2/ /

.1/

^ fn.2/ / D lim T t fn.2/

D lim lim T t .fk k!1 n!1

D lim lim T t .fk n!1 k!1

m-a.e.

n!1

Hence T t f is independent of the choice of the approximating sequence. For any f 2 L1 .XI m/, set T t f D T t f C  T t f  : We call ¹T t ; t > 0º or E conservative if T t 1 D 1 m-a.e. for any t > 0. Lemma 1.6.5. S T t 1 D 1 m-a.e. on Xc . If ¹T t ; t > 0º is recurrent, then it is conservative. If `1 ¹T` 1 < 1º D X m-a.e., then ¹T t ; t > 0º is transient. Proof. Since T t 1  1 m-a.e., we have, for a strictly positive function f 2 L1C \ L1 C,  Z .SN f; 1  T t 1/ D f;



N

0

Ts .1  T t 1/ds

 Z t Z D f; Ts 1ds  0

N Ct

N



 Z t  Ts 1ds  f; Ts 1ds  t .f; 1/: 0

Letting N ! 1, we get .Gf; 1  T t 1/  t.f; 1/ < 1. Since Gf D 1 m-a.e. on Xc , it follows that T t 1 D 1 m-a.e. on Xc . The last assertion also follows from this. Exercise 1.6.1. Consider the half space D D ¹x D .x 0 ; xd / W xd > 0º  Rd ; d  1. Prove that the Dirichlet form . 12 D; H01 .D// on L2 .D/ admits an associated transition density p t .x; y/ D .2 t/

 d2

²

 exp

jx  yj2  2t



  exp

jx  b y j2  2t

³ ;

x; y 2 D; b y D .y 0 ; yd /; (1.6.1) and that this Dirichlet form is transient. For any non-negative function g 2 L1 .XI m/ \ L1 .XI m/, we consider a perturbed form E g .u; v/ D E.u; v/ C .u; v/gm ; u; v 2 F ; .1:6:3/

1.6

Global properties of Markovian semigroups

57

R where .u; v/gm D X u.x/v.x/g.x/d m.x/: Since E.u; u/  E g .u; u/  .u; u/kgkL1 C E.u; u/; .E g ; F / is a Dirichlet space on L2 .XI m/. The notions associated with this Dirichlet space will be designated by the superscript g. Then the following lemma holds. Lemma 1.6.6. Let g be a non-negative function of L1 .XI m/ \ L1 .XI m/ such that g > 0 m-a.e. on Xc . Then the semigroup associated with .E g ; F / is transient. Moreover, G g g  1 m-a.e. on X and G g g D 1 m-a.e. on Xc . Proof. For any f 2 L2 .XI m/ and u 2 F , we have E˛ .G˛g f; u/ D E˛g .G˛g f; u/  .G˛g f; u/gm D .f  gG˛g f; u/;

.1:6:4/

and accordingly G˛g f D G˛ .f  gG˛g f /:

.1:6:5/

In particular, the inequality G˛g f

 G˛ f holds for any ˛ > 0 and f 2 L1C .XI m/. g In view of Lemma 1.6.1, we conclude that any T t -invariant set is also T t -invariant. g The inequality also implies the inclusion Xd  Xd up to an m-negligible set. On the other hand, .G˛g g; gG˛g g/  E˛g .G˛g g; G˛g g/ D .g; G˛g g/ 1=2  Z 1=2 Z .G˛g g/2 gd m gd m ;  X

and hence

Z

Z X

X

.G˛g g/2 gd m 

X

gd m < 1:

Consequently G g g < 1 m-a.e. on Xc . Therefore, for any strictly positive function h 2 L1 .XI m/, the function hIXd C gIXc is a strictly positive function of L1 .XI m/ such that G g .hIXd C gIXc / D IXd G g h C IXc G g g < 1 m-a.e. which g implies the transience of ¹T t ; t > 0º by Lemma 1.5.1. For the proof of the second assertion, we shall first prove that G g g  1 m-a.e. on X. Notice that the form .E; F / is also a Dirichlet form on L2 .XI .g C "/m/ for all " > 0. Since E.G"g ."f C fg/; v/ C .G"g ."f C fg/; v/.gC"/m D E g .G"g ."f C fg/; v/ C .G"g ."f C fg/; v/"m D ."f C fg; v/ D .f; v/.gC"/m ; G"g ."f C fg/ is equal to the 1-order resolvent of f with respect to the Dirichlet form .E; F / on L2 .XI .g C "/m/. Then, according to the Markov property of the

58

1 Basic theory of Dirichlet forms

resolvent, we have G"g ."f C fg/  1 m-a.e. for any f 2 L2 .XI m/ such that 0  f  1: It follows that G g g  1 m-a.e. Now the property that G g g D 1 on Xc  ¹g > 0º is a consequence of Lemma 1.6.3 and the equation G.g.1  G g g// D G g g < 1 which follows from (1.6.5). By using the equation (1.6.4), we get Corollary 1.6.1. Let g be a bounded m-integrable function such that g > 0 m-a.e. g on Xc and g D 0 m-a.e. on Xd . Then the functions un D G1=n g 2 F satisfy lim un .x/ D 1Xc .x/ m-a.e.

n!1

and

lim E.un ; un / D 0:

n!1

.1:6:6/

Recall the definition of the extended Dirichlet space .Fe ; E/ of .E; F /. Since un of Corollary 1.6.1 is an approximating sequence for 1Xc , we get Corollary 1.6.2. 1Xc 2 Fe and E.1Xc ; 1Xc / D 0. We are now in a position to give some characterizations of transience and recurrence. In the rest of this section, we assume (1.1.7). Theorem 1.6.2. The next conditions are equivalent to each other: (i) ¹T t ; t > 0º is transient. (ii) u D 0 m-a.e. whenever u 2 Fe and E.u; u/ D 0. (iii) Fe is a real Hilbert space with inner product E. Proof. The implication (i) ) (iii) follows from Theorem 1.5.1 and Lemma 1.5.5. (iii) ) (ii) is trivial. Suppose condition (ii) is satisfied, then IXc D 0 m-a.e. by Corollary 1.6.2, arriving at (i). Theorem 1.6.3. The next conditions are equivalent to each other: (i) ¹T t ; t > 0º is recurrent. (ii) There exists a sequence ¹un º  F satisfying limn!1 un D 1 m-a.e. and limn!1 E.un ; un / D 0: (iii) 1 2 Fe and E.1; 1/ D 0: Proof. (ii) and (iii) are equivalent. By virtue of Corollary 1.6.2, (i) implies (iii). On account of Theorem 1.6.2, (iii) implies the non-transience of ¹T t ; t > 0º, and consequently property (i) provided that ¹T t ; t > 0º was irreducible (Lemma 1.6.4). In order to derive (i) from (ii) in general, suppose that ¹T t ; t > 0º is nonrecurrent, namely, m.Xd / > 0, and that (ii) is fulfilled. By taking the unit contraction if necessary, we may suppose un ’s in (ii) satisfy 0  un  1. In view of

1.6

Global properties of Markovian semigroups

59

Theorem 1.6.1, we can define the reduced form .A; D/ of .E; F / to the invariant set Xd by D D ¹ujXd W u 2 F º and A.ujXd ; vjXd / D E.u1Xd ; v1Xd /, u; v 2 F . .A; D/ is a closed symmetric form on L2 .Xd I m/ associated with the strongly continuous Markovian semigroup ¹U t ; t > 0º on it defined by U t .ujXd / D .T t .u1Xd //jXd ; u 2 L2 .XI m/. Since ¹U t ; t > 0º is transient, we can find, as in the proof of Theorem 1.5.1, a strictly positive function g 2 L1 .Xd I m/ such that Z q un gd m  E.un 1Xd ; un 1Xd /: Xd

p The right-hand side being dominated byR E.un ; un / in view of Theorem 1.6.1, we let n ! 1 to arrive at the absurdity Xd gd m D 0:   Example 1.6.1. We consider the Dirichlet form 12 D; H 1 .D/ on L2 .D/ of Example 1.2.3 and suppose that D is a bounded domain in Rd . It is recurrent because 1 2 H 1 .D/ and D.1; 1/ D 0. Let He1 .D/ be its extended Dirichlet space. As was observed in Example 1.5.3, He1 .D/  G .D/ where G .D/ is the space introduced in Example 1.2.3. Denote by C0 .D/ (resp. C01 .D/) the space of restriction to D of functions in C0 .Rd / (resp. C01 .Rd /). G .D/ \ C0 .D/ is not necessarily D-dense in G .D/. For instance, if we let D D ¹.r;  / W 1 < r < 2; 0 <  < 2 º; D1 D ¹.r;  / W 1 < r < 2; 0   < 2 º; then the function u.r;  / D  is in G .D/ but not in G .D1 / because it violates the property .G :3/ on D1 . Hence it can not be D-approximated by functions in C01 .D/ D C01 .D 1 /. Suppose that a bounded domain D is of the class C in the sense that any x 2 @D has a neighbourhood U such that D \ U D ¹.x1 ; x2 ; : : : ; xd / W xd > F .x1 ; x2 ; : : : ; xd 1 /º \ U in some coordinate .x1 ; x2 ; : : : ; xd / and with a continuous function F . Then C01 .D/ is known to be dense both in H 1 .D/ and G .D/.20 Hence . 12 D; H 1 .D// can be viewed as a regular Dirichlet form on L2 .D/ (rather than on L2 .D/). Furthermore21 He1 .D/ D G .D/: .1:6:7/ V. G. Maz0 ja [1]. (1.6.7) holds for any domain D  Rd of finite Lebesgue measure (cf. Chen–Fukushima [1; Cor.2.2.15]). 20 Cf.

21 Actually

60

1 Basic theory of Dirichlet forms

Example 1.6.2. We are concerned with the Sobolev spaces on two dimensional domains. Let us first consider the Dirichlet form . 12 D; H 1 .R2 // on L2 .R2 /. We have seen in Example 1.5.2 that this form is R non-transient. Since the associated transition function is given by p t f .x/ D R2 q t .x  y/f .y/dy with a strictly positive density 1  jxj2 q t .x/ D e 2t ; 2 t the present Dirichlet form is irreducible and accordingly recurrent. Let .He1 .R2 /; E/ be the corresponding extended Dirichlet space. Then we have22   1 1 2 2 .He .R /; E/ D G .R /; D .1:6:8/ 2 where G .R2 / is the space introduced in Example 1.2.3. We know by Theorem 1.6.3 that 1 2 He1 .R2 / and E.1; 1/ D 0. If we can assert more strongly that (a) E.u; u/ D 0 if and only if u is constant, (b) the quotient space HP e1 .R2 / of He1 .R2 / by constant functions is a Hilbert space with respect to E, then the same arguments as in Example 1.5.3 lead us to the identification HP e1 .Rd / D GP .Rd / and consequently (1.6.8). We shall get (a) and (b) by proving the following counterpart of (1.5.8): there exists an integrable bounded and strictly positive function g and a continuous linear functional L on L1 .R2 I g  dx/ such that He1 .R2 /  L1 .R2 I g  dx/ and Z p ju.x/  L.u/jg.x/dx  E.u; u/; u 2 He1 .R2 /: .1:6:9/ R2

Clearly (1.6.9) implies (a) and (b). To show (1.6.9), we consider the logarithmic R potential Vf .x/ D R2 v.x  y/f .y/dy of functions f where v.x/ D

1 1 ; log jxj

x 2 R2 :

We take a strictly positive bounded integrable function g with Z Z jv.x  y/j g.x/g.y/dxdy < 1; R2

R2

3 2 Re.g. g.x/ D 1=.1 C jxj /. We also fix a non-negative function ' 2 C0 .R / with R2 'dx D 1 and set Z u'dx; u 2 L1 .R2 I g  dx/: L.u/ D R2

22 (1.6.8)

can be obtained more directly (cf. Th. 2.2.13 or §6.5 .5ı / of Chen–Fukushima [1]).

1.6

Let f .x/ D g.x/  relation23

R R2

Global properties of Markovian semigroups

gdx  '.x/ so that Z

v.x/ D lim

T !1 0

T

R R2

61

f dx D 0. Observing the

  1 1 2t q t .x/  dt; e 2 t

we have that lim ST f .x/ D Vf .x/;

T !1

where ST f .x/ D

RT 0

x 2 R2 ;

lim .f; ST f / D .f; Vf / < 1;

T !1

p t f .x/dt . In view of Lemma 1.5.3, ST f 2 H 1 .R2 / and

E.ST f; u/ D .f  pT f; u/;

8u 2 H 1 .R2 /:

Then E.ST f  ST 0 f; ST f  ST 0 f / D .f; 2ST CT 0 f  S2T f  S2T 0 f / ! 0; T; T 0 ! 1, and hence Vf 2 Fe . Since .pT f; f / D .pT =2 f; pT =2 f / decreases to zero as T ! 1, we further get E.Vf; u/ D .f; u/; 8u 2 H 1 .R2 /:   R Now .f; u/ D g  R2 gdx  '; u D .u  L.u/; g/. R Therefore if we replace g .u and f by g1 D g  sgn  L.u// and f1 D g1  R2 g1 dx  ' respectively, we get Z p p ju  L.u/j  gdx D .u  L.u/; g1 / D E.Vf1 ; u/  .f1 ; Vf1 / E.u; u/: But .f1 ; Vf1 / is dominated by “ “ jv.x  y/jg.x/'.y/dxdy jv.x  y/jg.x/g.y/dxdy C 2kgk1 “ C

kgk21

jv.x  y/j'.x/'.y/dxdy D  2 ;

which is finite and independent of u. Hence, replacing g with g=, we arrive at the desired inequality (1.6.9) for u 2 H 1 .R2 /, which is readily extended to u 2 He1 .R2 /. In Corollary 4.8.1 of §4.8, we shall derive the inequality (1.6.9) for a much more general irreducible recurrent Dirichlet form .E; F /: Next let D be the upper half plane: D D ¹.x1 ; x2 / W x2 > 0º: 23 Cf.

S. C. Port–C. J. Stone [1].

62

1 Basic theory of Dirichlet forms

Just as in the latter half of Example 1.5.3, any function u 2 G .D/ admits (after a modification on a set of zero Lebesgue measure) an extension to a function of G .R2 / by reflection. Consequently we conclude from (1.6.8) the analogous property of the extended Dirichlet space .He1 .D/; E/ of .H 1 .D/; 12 D/24 .He1 .D/;

  1 E/ D G .D/; D : 2

.1:6:10/

Further C01 .D/, the restrictions to D of functions in C01 .R2 /, is dense in H 1 .D/ and hence . 12 D; H 1 .D// is a regular Dirichlet form on L2 .D/. (1.6.10) particularly implies the recurrence of . 12 D; H 1 .D// because 1 2 He1 .D/ and E.1; 1/ D 0. To the contrary, the smaller Dirichlet form . 12 D; H01 .D// is transient by Exercise 1.6.1. Therefore the relation (1.5.23) in Example 1.5.3 is valid and we get from (1.6.10) 1 .D/ ˚ HP .D/; HP e1 .D/ D H0;e

.1:6:11/

where HP e1 .D/ (resp. HP .D/) is the quotient space of He1 .D/ (resp. H .D/) by 1 .D/ is the extended Dirichlet space of the space of constant functions and H0;e 1 1 . 2 D; H0 .D//. Exercise 1.6.2. Show that He1 .R1 / D ¹u W u is absolutely continuous on R1 and D.u; u/ < 1º: The next comparison theorem is an immediate consequence of Theorem 1.5.1 and Theorem 1.6.3. .i /

Theorem 1.6.4. Let ¹T t ; t > 0º be strongly continuous Markovian semigroups on L2 .XI m.i / / with associated Dirichlet spaces .E .i / ; F .i / /; i D 1; 2. We assume that m.1/ and m.2/ are mutually absolutely continuous and that F .1/ F .2/ ;

E .1/ .u; u/  C E .2/ .u; u/;

u 2 F .2/ ;

for some positive constant C . .1/

.2/

.2/

.1/

(i) If ¹T t ; t > 0º is transient, then so is ¹T t ; t > 0º. (ii) If ¹T t ; t > 0º is recurrent, then so is ¹T t ; t > 0º. The next theorem is a variant of Theorem 1.6.3. 24 Actually

(1.6.10) holds for any domain D  R2 (Cf. Chen–Fukushima[1; §6.5 .5ı /]).

1.6

Global properties of Markovian semigroups

63

Theorem 1.6.5. The following conditions are equivalent to each other: (i) ¹T t ; t > 0º is recurrent. (ii) There exists a sequence ¹un º  F satisfying 0  un  1;

lim un D 1 m-a.e.

n!1

.1:6:12/

such that lim E.un ; v/ D 0

n!1

.1:6:13/

holds for any v 2 Fe . (iii) There exists a sequence ¹un º  F satisfying .1:6:12/ such that .1:6:13/ holds g for bounded g 2 L1 .XI m/ with R v D G f 2 Fe for some non-negative 1 X gd m > 0 and strictly positive f 2 L .XI m/. Proof. Obviously the third condition of Theorem 1.6.3 implies (ii), from which follows (iii) trivially. Suppose that (iii) is fulfilled. Since E.un ; G g f / D .un ; f /  .un ; G g f /gm D .un  G g .gun /; f /; we have .1  G g g; f / D 0 and hence G g g D 1 m-a.e. In view of Lemma 1.6.6 and Theorem 1.5.3, we conclude that 1 D G g g 2 Feg and E.1; 1/ D 0, which implies the third condition of Theorem 1.6.3 because Feg  Fe . We now state conservativeness criteria quite analogous to the preceding recurrence criteria. Theorem 1.6.6. The following conditions are equivalent to each other: (i) ¹T t ; t > 0º is conservative. (ii) There exists a sequence ¹un º  F satisfying .1:6:12/ such that .1:6:13/ holds for any v 2 F \ L1 .XI m/. (iii) There exists a sequence ¹un º of functions of F satisfying .1:6:12/ such that .1:6:13/ holds for some v D G˛ f; ˛ > 0 with f 2 L1 .XI m/ \ L2 .XI m/; f > 0 m-a.e. Proof. Suppose that ¹T t ; t > 0º is conservative. By taking a sequence ¹fn º of functions in L2 .XI m/ satisfying 0  fn  1; fn  fnC1 ; fn " 1 m-a.e., put un D G1 fn . Then un 2 F ; un " 1 m-a.e. and hence, for any v 2 F \ L1 .XI m/, lim E.un ; v/ D lim .fn  G1 fn ; v/ D 0:

n!1

n!1

This proves (i) ) (ii). (ii) ) (iii) is obvious.

64

1 Basic theory of Dirichlet forms

Suppose that .1:6:13/ holds for some v D G˛ f; ˛ > 0; f 2 L1 .XI m/ \ f > 0 and un 2 F satisfies .1:6:12/. By letting n ! 1 in the equality E.un ; G˛ f / D .un  ˛G˛ un ; f /; L2 .XI m/,

we get .1  ˛G˛ 1; f / D 0. This proves (iii) ) (i). If ¹T t ; t > 0º is irreducible, then the function f appearing in the statement (iii) of the preceding two theorems can be any non-negative function such that 0 < .1; f / < 1. We have employed in (1.6.3) a perturbed Dirichlet form E g . This simple perturbation is useful in reducing the general case to transient ones which are easier to be handled. Here we state a basic lemma involving E g , which will be utilized in later chapters to investigate properties of (not necessarily transient) extended Dirichlet space Fe . Let us consider a collection of functions Z ° ± K D g W g is bounded, strictly positive m-a.e., gd m  1 : .1:6:14/ X

For g 2 K, we let E g .u; v/ D E.u; v/ C .u; v/gm ;

u; v 2 F :

As we have seen in the paragraph preceding to Lemma 1.6.6, .F ; E g / is a Dirichlet space on L2 .XI m/. It is transient and possessing just g as its reference function in the sense of §1.5 because sZ Z p u2 gd m  E g .u; u/; u 2 F : jujgd m  Denote by .Feg ; E g / the (transient) extended Dirichlet space of .F ; E g /. Lemma 1.6.7. For any u 2 Fe , there exists a g 2 K such that u 2 Feg . sequence of u 2 Fe . For any bounded Proof. Let ¹un º be an approximating R strictly positive function f with X f d m  1, let g.x/ D .supn1 u2n .x/ _ 1/1  f R .x/. Obviously, 2g is an m-a.e. positive bounded function dominated by f and g X .un .x/  u.x// g.x/m.dx/ ! 0 as n ! 1. Therefore u is an element of Fe g with an E -approximating sequence un 2 F ; n D 1; 2; : : : : Since Feg  Fe and E g dominates E on Feg , we obtain the following in view of Theorem 1.5.2 (ii), Theorem 1.5.3 (ı) and Corollary 1.5.1:

1.6

Global properties of Markovian semigroups

65

Corollary 1.6.3. Every normal contraction operates on .Fe ; E/. Assertions (i), (iii), (iv) and the second assertion of (v) in Theorem 1:4:2 remain valid with DŒE and E1 being replaced by Fe and E respectively. Further the product of bounded functions in Fe is again in Fe .

Chapter 2

Potential theory for Dirichlet forms

The first three sections of this chapter are devoted to a presentation of the potential theory for regular Dirichlet forms mainly due to Beurling and Deny. Its connections with classical potential theory based on the Riesz kernel and the logarithmic kernel are stated in the examples of §2.2. BLD functions of potential type are explored in examples of §2.1, §2.2 and §2.3. Fundamental notions in the potential theory (sets of capacity zero, quasi continuity of functions, equilibrium potentials, etc.) shall be interpreted probabilistically in Chapter 4. Notions of regular nests (§2.1) and smooth measures (§2.2) shall play important roles in Chapter 7 and Chapter 5, respectively. Beurling– Deny’s theorem on spectral synthesis (§2.3) shall be utilized in §4.4 and §5.4. Throughout Chapter 2 we fix a pair .X; m/ satisfying (1.1.7) and consider a regular Dirichlet form E on L2 .XI m/.1 As in §1.5, we denoted DŒE by F . Although we present the potential theory relevant to the Hilbert space .F ; E1 /, an analogous presentation is possible in terms of the extended Dirichlet space .Fe ; E/ provided that E is transient. The latter will be formulated and examined in the second halves of §2.1, §2.2 and §2.3. §2.4 presents a general relationship between capacitary estimates and Poincaré type and Sobolev type inequalities. The relation is then applied to symmetric stable semigroups using capacities based on the Bessel and Riesz potential spaces.

2.1

Capacity and quasi continuity

Denote by O the family of all open subsets of X. For A 2 O we define LA D ¹u 2 F W u  1 m-a.e. on Aº; ´ infu2LA E1 .u; u/; LA ¤ ; Cap.A/ D 1 LA D ;;

(2.1.1) (2.1.2)

and for any set A  X we let Cap.A/ D

inf

B2O;AB

Cap.B/:

.2:1:3/

1 As for the possible milder topological assumptions on X , see Exercise 2.1.1 at the end of §2.1 and a reduction theorem (Theorem A.4.3) in the Appendix A.4.

2.1

Capacity and quasi continuity

67

We call this 1-capacity of A or simply the capacity of A. Let O0 D ¹A 2 O I LA ¤ º. Lemma 2.1.1. (i) For each A 2 O0 , there exists a unique element eA 2 LA such that E1 .eA ; eA / D Cap.A/: .2:1:4/ (ii) 0  eA  1 m-a.e. and eA D 1 m-a.e. on A. (iii) eA is a unique element of F satisfying eA D 1 m-a.e. on A and E1 .eA ; v/  0; 8v 2 F ; v  0 m-a.e. on A. (iv) v 2 F ; v D 1 m-a.e. on A ) E1 .eA ; v/ D Cap.A/: (v) A; B 2 O0 ; A  B ) eA  eB m-a.e. Proof. (i) Clearly LA is a convex closed subset of .F ; E1 /. By making use of the equality     1 1 uv uv uCv uCv ; C E1 ; D E1 .u; u/ C E1 .v; v/ E1 2 2 2 2 2 2 we can see that any minimizing sequence un 2 LA .limn!1 E1 .un ; un / D Cap.A// is E1 -convergent to an element eA 2 LA satisfying (2.1.4) and that such an eA is unique. (ii) By .E:4/0 ; u D .0 _ eA / ^ 1 2 LA and E1 .u; u/  E1 .eA ; eA / D Cap.A/. Hence u D eA . (iii) If v has the stated property, then eA C "v 2 LA and E1 .eA C "v; eA C "v/  E1 .eA ; eA / for any " > 0. From this we get E1 .eA ; v/  0. Conversely, suppose u 2 F satisfies the conditions in (iii), then u 2 LA and w  u  0 m-a.e. on A for any w 2 LA . Hence E1 .w; w/ D E1 .u C .w  u/; u C .w  u//  E1 .u; u/; 8w 2 LA , proving u D eA . (iv) Immediate from (iii). (v) Notice that E1 .uC ; u /  0 for any u 2 F . Since eA eA ^eB D .eA eB /C vanishes m-a.e. on A, E1 .eA  eA ^ eB ; eA  eA ^ eB / D E1 .eA ^ eB ; .eA  eB /C / D E1 ..eA  eB / ; .eA  eB /C /  E1 .eB ; .eA  eB /C /  0 and hence eA D eA ^ eB m-a.e. The capacity defined by (2.1.2) has the following properties on the family O.

68

2 Potential theory for Dirichlet forms

Lemma 2.1.2. (i) A; B 2 O; A  B ) Cap.A/  Cap.B/. (ii) Cap.A [ B/ C Cap.A \ B/  Cap.A/ C Cap.B/; A; B 2 O. S  (iii) An 2 O; An " ) Cap n An D supn Cap .An /. Proof. (i) is clear. For A; B 2 O0 , Cap.A [ B/ C Cap.A \ B/  E1 .eA _ eB ; eA _ eB / C E1 .eA ^ eB ; eA ^ eB / 1 1 D E1 .eA C eB ; eA C eB / C E1 .jeA  eB j; jeA  eB j/ 2 2  E1 .eA ; eA / C E1 .eB ; eB / D Cap.A/ C Cap.B/; which proves (ii). It suffices to prove (iii) when supn Cap.An / is finite. By Lemma 2.1.1, we then have for n > m; E1 .eAn  eAm ; eAn  eAm / D Cap.An /  Cap.Am /. Therefore S eAn is E1 -convergent to some u 2 F . Clearly u D 1 m-a.e. on A D An . If v 2 F is non-negative m-a.e. on A; E1 .u; v/ D limn!1 E1 .eAn ; v/  0. Hence u D eA by Lemma 2.1.1 and supn Cap.An / D limn!1 E1 .eAn ; eAn / D E1 .u; u/ D Cap.A/. As a consequence of Lemma 2.1.2 and Theorem A.1.2 we can obtain the next theorem. Theorem 2.1.1. The capacity defined by .2:1:2/ and .2:1:3/ is a Choquet capacity, i.e., (a) A  B ) Cap.A/  Cap.B/. S (b) An " ) Cap. n An / D supn Cap.An /. T (c) An compact, An # ) Cap. n An / D infn Cap.An /. By virtue of Corollary A.1.1, it holds that for any Borel set A Cap.A/ D

sup

Cap.K/:

.2:1:6/

K compact; KA

The present notion of capacity enables us to think of exceptional sets finer than m-negligible sets. In fact, the inequality m.A/  Cap.A/ for A 2 O following from the definition (2.1.2) implies that any set of zero capacity is m-negligible. Let A be a subset of X. A statement depending on x 2 A is said to hold q.e. on A if there exists a set N  A of zero capacity such that the statement is true for every x 2 A n N . “q.e.” is an abbreviation of “quasi-everywhere”.

2.1

Capacity and quasi continuity

69

Let X D X [  be the one-point compactification of X. When X is already compact,  is regarded as an isolated point. For any subset A  X; A [  is endowed with the relative topology as a subspace of X . Any function u on A is always extended to a function on A [  by setting u./ D 0. In particular, any function belonging to C1 .X/ is regarded as a continuous function on X . Let u be an extended real valued function defined q.e. on X. We call u quasi continuous2 if there exists for any " > 0 an open set G  X such that Cap.G/ < " and ujX nG is finite continuous. Here ujX nG denotes the restriction of u to X n G. If we require a stronger condition by replacing ujX nG in the above definition with ujX nG , then we say that u is quasi continuous in the restricted sense. A sequence ¹Fk º of closed sets such that Fk " and Cap.X n Fk / # 0; k " 1, is called a nest on X. A nest ¹Fk º is said to be m-regular if for each k; supp.IFk m/ D Fk , i.e., m.U.x/ \ Fk / ¤ 0 for any x 2 Fk and any open neighbourhood U.x/ of x. For a closed set F , we let F 0 D suppŒIF  m. Since F 0 is the smallest we have that F 0  F and closed set whose R complement is IF 0 m-negligible, 0 0 m.G n G/ D G 0 IF  d m D 0 for G D X n F ; G D X n F . In particular Cap.G 0 / D Cap.G/

.2:1:7/

and we get the following: Lemma 2.1.3. For a given nest ¹Fk º on X, ¹Fk0 º is an m-regular nest. Given a nest ¹Fk º on X, let C.¹Fk º/ D ¹u W ujFk is continuous for each kº C1 .¹Fk º/ D ¹u W ujFk [ is continuous for each kº:

(2.1.8) (2.1.9)

Clearly C1 .¹Fk º/  C.¹Fk º/; C1 .X/  C1 .¹Fk º/ and C.X/  C.¹Fk º/. Theorem 2.1.2. (i) Let S be a countable family of quasi continuous functions (resp. quasi continuous functions in the restricted sense) on X. Then there exists a regular nest ¹Fk º on X such that S  C.¹Fk º/ .resp. S  C1 .¹Fk º//. nest on X and u belong to C.¹Fk º/. If u  0 m-a.e., (ii) Let ¹Fk º be a regular S then u.x/  0; 8x 2 1 kD1 Fk . e k º of closed sets Proof. (i) If u is quasi continuous, there exists a sequence ¹F Sk e k / < 1=k and uj e such that Cap.X n F lD1 F l ; k D e F k is continuous. Let Fk D 1; 2; : : : ; then ¹Fk º is a nest and u 2 C.¹Fk º/. 2 In formulating the quasi continuity in this book, we only consider the numerical function which is finite q.e.

70

2 Potential theory for Dirichlet forms

Given a family S D ¹ul º of quasi continuous functions ul , choose for each l .l/ .l/ .l/ a nest ¹Fk º such that ul 2 C.¹Fk º/; Cap.X n Fk / < .1=2l /.1=k/. Let T1 P .l/ .l/ Fk D lD1 Fk . Since Cap.X n Fk /  1 lD1 Cap.X n Fk /  1=k; ¹Fk º is a nest again. Clearly S  C.¹Fk º/. It suffices now to regularize ¹Fk º according to Lemma 2.1.3. (ii) Suppose u.x/ < 0 for some x 2 Fk , then there is a neighbourhood U.x/ such that u.y/ < 0; 8y 2 U.x/ \ Fk , on account of the continuity of ujFk . Since ¹Fk º is m-regular, m.U.x/ \ Fk / > 0, contradicting the assumption that u  0 m-a.e. In particular, the preceding theorem implies the following; if u is quasi continuous and u  0 m-a.e., then u  0 q.e. This assertion can be localized to an arbitrary open set G  X. In fact, thinking of G instead of X, the notion of quasi continuity of a function on G can be defined; furthermore, we can make assertions analogous to Lemma 2.1.3 and Theorem 2.1.2. In particular, we have Lemma 2.1.4. Let G be an open set of X and u be quasi continuous on G. If u  0 m-a.e. on G, then u  0 q.e. on G. Along with the quasi continuity of numerical functions, we can introduce the quasi notions for subsets of X. A set E  X is called quasi open if for any " > 0 there exists an open set G containing E with Cap.G n E/ < ". A quasi closed set is by definition the complement of a quasi open set. A set E is quasi open (resp. quasi closed) if and only if there exists a nest ¹Fk º such that Fk \ E is a relatively open subset of Fk (resp. a closed set) for each k. Indeed, if E is quasi closed, then, for any " > 0, there is an open set ! such that Cap.!/ < " and E  ! is closed. An extended real valued function u is quasi continuous if and only if u is finite q.e. and u1 .I / is quasi open for any open set I  R. The “only if” S1part is obvious. To see the “if” part, choose a nest ¹Fk º such that u is finite on kD1 Fk and both ¹u  rº \ Fk and ¹u  rº \ Fk are closed for every rational r and k D 1; 2; : : : : Then u is quasi continuous with the associated nest ¹Fk º. Lemma 2.1.5. Let u be a quasi continuous function on X and E be a quasi open subset of X. If u  0 m-a.e. on E, then u  0 q.e. on E. Proof. Using Lemma 2.1.3, construct a regular nest such that u 2 C.¹Fk º/ and E \ FSk is a relatively open subset of Fk for each k. Then u.x/  0; 8x 2 E \. 1 kD1 Fk /.

2.1

Capacity and quasi continuity

71

So far in this section we have not used the regularity of the Dirichlet form E. We now state three theorems based on this property of E. Given two functions u and v; v is said to be a quasi continuous modification of u (in the restricted sense) if v is quasi continuous (in the restricted sense) and v D u m-a.e. In this case, we designate v by e u. Theorem 2.1.3. Each u 2 F admits a quasi continuous modification e u in the restricted sense. Proof. First we establish the inequality Cap.¹x 2 X W ju.x/j > º/ 

1 E1 .u; u/; 2

 > 0; u 2 F \ C.X/: .2:1:10/

Since G D ¹x 2 X W ju.x/j > º 2 O and juj= 2 LG for any  > 0 and u 2 F \ C.X/, we have Cap.G/  .1=2 /E1 .juj; juj/  .1=2 /E1 .u; u/, proving (2.1.10). In view of the regularity of E, any u 2 F can be approximated with respect to the E1 -metric by some un 2 F \ C0 .X/. We may assume E1 .ukC1  uk ; ukC1  uk / < 23k by selecting a subsequence if necessary. Then Cap.Gk /  2k by (2.1.10), where Gk D ¹x 2 X W jukC1 .x/  uk .x/jP> 2k º. Let Fk D T 1 1 c lDk Gl . Clearly ¹Fk º is a nest and jun .x/  um .x/j  DN C1 juC1 .x/  N for any x 2 Fk and any n; m > N  k. This means, for u .x/j  2 each k; un jFk [ (by setting un ./SD 0) are uniformly convergent as n ! 1. Let e u.x/ D limn!1 un .x/; x 2 1 u 2 C1 .¹Fk º/ and u D e u kD1 Fk I then e m-a.e. e the set of all quasi continuous functions belonging to F . By the Denote by F e (in the sense of above theorem and Lemma 2.1.4, the equivalence classes of F q.e.) are identical with the equivalence classes of F (in the sense of m-a.e.). e. Lemma 2.1.6. Inequality .2:1:10/ holds for any u 2 F e , consider un 2 F \ C0 .X/ and its limit e Proof. For u 2 F u which appeared in the proof of Theorem 2.1.3. Since u D e u q.e., there is for any " > 0 an open set G such that Cap.G/ < " and un converges to u uniformly on X nG. Hence, for any ; "1 with  > "1 > 0; ¹x 2 X W ju.x/j > º  ¹x 2 X W jun .x/j >   "1 º [ G if n is sufficiently large. Then, by (2.1.10), Cap.¹x 2 X W ju.x/j > º/  E1 .un ;un / C ". By letting n ! 1; "1 ! 0 and then " ! 0, we get the inequality ."1 /2 e. (2.1.10) for u 2 F

72

2 Potential theory for Dirichlet forms

e constitutes an E1 -Cauchy sequence, then there Theorem 2.1.4. (i) If un 2 F e such that un ! u q.e. and ¹un º is are a subsequence nk and a u 2 F k E1 -convergent to u. (ii) If un 2 F constitutes an E1 -Cauchy sequence and if some modifications e of un converge to a function e e and ¹un º is e un 2 F u q.e., then e u 2 F E1 -convergent to e u. e and E1 .un  um ; Proof. (ii) follows from (i) at once. Suppose that un 2 F un  um / ! 0. By making use of Lemma 2.1.6, we can see exactly as in the proof of Theorem 2.1.3 that there exist a subsequence nl and a decreasing sequence of sets ¹Ek º such that limk!1 Cap.Ek / D 0 and unl converges uniformly on each X n Ek as nl ! 1. Let u be the limit function. Choose for any " > 0 an open set G1 such that Cap.G1 / < "=2 and G1 Ek for sufficiently large k. Also choose an open set G2 such that Cap.G2 / < "=2 and unl jX nG2 is continuous for every nl , which is possible in view of Theorem 2.1.2 (i). Let G D G1 [ G2 . Then Cap.G/ < " and unl converges to u uniformly on e . Clearly ¹un º is also E1 -convergent to u. X n G. Hence u 2 F Finally we can extend Lemma 2.1.1 as follows. For an arbitrary set B  X, we let LB D ¹u 2 F W e u  1 q.e. on Bº:

.2:1:11/

When B is open, LB coincides with the space (2.1.1) in view of Lemma 2.1.4. Theorem 2.1.5. Fix an arbitrary set B  X. (i) Cap.B/ D infu2LB E1 .u; u/. (ii) If LB ¤ , then there exists a unique element eB in LB minimizing E1 .u; u/. eB satisfies Cap.B/ D E1 .eB ; eB /: .2:1:12/ (iii) 0  eB  1 m-a.e. and e eB D 1 q.e. on B. eB D 1 q.e. on B and E1 .eB ; v/  0 (iv) eB is a unique element of F satisfying e for any v 2 F with e v  0 q.e. on B. Proof. We first prove (ii). If LB is non-empty, it is a closed convex subset of .F ; E1 / on account of Theorem 2.1.4. As in the proof of Lemma 2.1.1, we can find a unique element eB 2 LB such that E1 .eB ; eB /  E1 .u; u/; 8u 2 LB . For any " > 0, there exists A 2 O0 such that B  A, Cap.B/ > Cap.A/  ". Since eA of Lemma 2.1.1 belongs to LB by Lemma 2.1.4, Cap.A/ D E1 .eA ; eA /  E1 .eB ; eB / proving the inequality “” in (2.1.12).

2.1

Capacity and quasi continuity

73

To get the converse inequality, fix a quasi continuous modificatione eB of eB . For eB jX nA" is continuous any " > 0, choose an open set A" such that Cap.A" / < "; e and e eB  1 on B \ .X n A" /. Denote by e" the function eA" of Lemma 2.1.1 for the open set A" . Now the set G" D ¹x 2 X n A" W e eB > 1  "º [ A" is open and B  G" . Moreover, eB C e"  1  " m-a.e. on G" . Therefore Cap.B/  Cap.G" /  .1  "/2 E1 .eB C e" ; eB C e" / p p  .1  "/2 ¹ E1 .eB ; eB / C E1 .e" ; e" /º2 p p  .1  "/2 ¹ E1 .eB ; eB / C "º2 : By letting " # 0, we arrive at (2.1.12). (i) is an immediate consequence of (ii). (iii) and (iv) also follow from (ii) as in the proof of Lemma 2.1.1. The function eB in the above theorem is said to be 1-equilibrium potential of the set B. When B 2 O0 ; eB coincides with the function appearing in Lemma 2.1.1. As an immediate consequence of Lemma 2.1.5 and Theorem 2.1.5, we get Lemma 2.1.7. (i) .2:1:2/ extends from open sets to quasi open sets: if E is quasi open, then Cap.E/ D inf¹E1 .u; u/ W u 2 F ; u  1 m-a.e. on Eº: (ii) Any m-negligible quasi open set is of zero capacity. We could have employed the metric E˛ for positive ˛ instead of E1 in the definition (2.1.2) of capacity. However, it would make no difference as far as the notions of “q.e.” and “quasi continuity” are concerned because of the inequality .1 ^ ˛/E1 .u; u/  E˛ .u; u/  .1 _ ˛/E1 .u; u/; u 2 F . We note that we have the notion of ˛-equilibrium potential of a set using the metric E˛ . Suppose now that the Dirichlet space .E;F / is transient in the sense of §1.5. .E;F / is then completed to the extended Dirichlet space .Fe ;E/ by Theorem 1.5.2. Fe is a real Hilbert space with inner product E and Z p ju.x/jg.x/m.dx/  E.u; u/; 8u 2 Fe ; .2:1:13/ X

g being a reference function of .E; F / (see §1.5).

74

2 Potential theory for Dirichlet forms

Let us define the 0-order capacity Cap.0/ .A/ by replacing F and E1 in (2.1.1) and (2.1.2) with Fe and E respectively. Clearly Cap.0/ .A/  Cap.A/;

A  X;

.2:1:14/

but Cap.0/ .A/ D 0 still implies m.A/ D 0 on account of (2.1.13). Moreover, by making use of the above mentioned properties of .Fe ; E/ we can readily restate all the statements established in this section in terms of .Fe ; E/ and Cap.0/ . In .0/

particular, we have the (0-order) equilibrium potential eB corresponding to the modified version of Theorem 2.1.5. Lemma 2.1.8. Let ¹An º be a decreasing sequence of relatively compact open sets. Then Cap.0/ .An / # 0 if and only if Cap.An / # 0. Proof. The “if” part is trivial from (2.1.14). Suppose that Cap.0/ .An / # 0. Then by the modified version of (2.1.4) and the inequality (2.1.13), .0/

eAn ! 0 m-a.e.

.2:1:15/

by selecting a subsequence if necessary. Since A1 is compact and E is regular, there is a non-negative function h 2 .0/ F \ C0 .X/ such that h.x/  1; x 2 A1 . We let en0 D eAn ^ h. Then en0 2 L2 .XI m/ \ Fe D F by Theorem 1.5.2 (ii). It is easy to see that E1 .en0 ; en0 / is uniformly bounded by Cap.0/ .A1 / C E1 .h; h/. Hence the Cesàro means fn of some subsequence of ¹en0 º are E1 -convergent. On the other hand, (2.1.15) implies that en0 converges to zero m-a.e. and so does fn . Therefore, limn!1 E1 .fn ; fn / D 0. We now conclude Cap.An / ! 0 because fn 2 LAn and Cap.An /  E1 .fn ; fn /. Theorem 2.1.6. (i) Cap.0/ .A/ D 0 if and only if Cap.A/ D 0 for A  X. (ii) A function is quasi continuous with respect to Cap.0/ if and only if it is quasi continuous with respect to Cap. Proof. (i) is clear from Lemma 2.1.8. The “if” part of (ii) is also clear from (2.1.14). Let ¹Gn º be relatively compact open sets which converge to X. If u is quasi continuous with respect to Cap.0/ , then it is also quasi continuous on each Gn with respect to Cap by virtue of Lemma 2.1.8. Then it is easy to see that u is quasi continuous on X . In particular, each element of the transient extended Dirichlet space Fe admits a quasi continuous modification (with respect to Cap). We can prove that this is

2.1

Capacity and quasi continuity

75

still true for the general (not necessarily transient) extended Dirichlet space Fe by using Lemma 1.6.7. Let us turn back to a general (not necessarily transient) regular Dirichlet space .E; F / on L2 .XI m/ and consider its extended Dirichlet space .Fe ; E/. Theorem 2.1.7. Any u 2 Fe admits a quasi continuous modification e u. Moreover, for any core C of .E; F /, there exist un 2 C; n D 1; 2; : : : ; which are E-Cauchy and converge to e u q.e. as n ! 1. Proof. Let u 2 Fe and g be a function in the class K satisfying the condition of Lemma 1.6.7. Since the Dirichlet space .E g ; F / is transient and the metric E1g is equivalent to E1 on F , the stated properties follow from the preceding argument applied to the transient extended Dirichlet space .Feg ; E g /. Example 2.1.1 (BLD functions of potential type).3 In Example 1.5.3, we have  1 .D/ of the Dirichlet form 12 D; H01 .D/ . treated the extended Dirichlet space H0;e e 1 .D/ of all quasi continuous modifications of We may consider the space H 0;e 1 elements of H0;e .D/ in accordance with Theorem 2.1.7. We first suppose that H01 .D/ is transient. This is always true when d  3 (see Example 2.3.2 for e 1 .D/ is called a BLD .Beppo–Levi– d D 2). In this case, each element of H 0;e Deny/ function of potential type. In view of the 0-order version of Theorem 2.1.4, e 1 .D/ if and only if there is a sequence un 2 C 1 .D/ we can then see that u 2 H 0;e 0 such that ¹un º constitutes a Cauchy sequence with respect to the Dirichlet integral and un converges to u q.e. on D. 1 In the general (not necessarily transient) case, H0;e .D/ is a subspace of G .D/ 1 .D/ admits of (1.2.14) as was seen in Example 1.5.3. Hence each function of H0;e for each i (1  i  n) a version possessing property (G :3) stated after (1.2.14). However, as will be seen in Example 2.2.2 and Example 2.2.3, each element of e 1 .D/ has the stronger property that it is absolutely continuous on almost all H 0;e straight lines parallel to each axis no matter how the axes are taken. Example 2.1.2 (one-dimensional case). Let I D .r1 ; r2 / be a one-dimensional interval and let m and k be positive Radon measures on I with suppŒm D I . We consider the regular Dirichlet space .F 0 ; E/ on L2 .I I m/ introduced in Example 1.2.2 by (1.2.7) and (1.2.9). We saw in Example 1.2.2 that any E1 -Cauchy sequence un 2 F 0 is uniformly convergent on each compact subinterval of I . Therefore there exists for each y 2 I a unique g10 .  ; y/ 2 F 0 such that E1 .g10 .  ; y/; v/ D v.y/; 3 Cf.

J. Deny–J. L. Lions [1] and J. L. Doob [2].

v 2 F 0:

.2:1:16/

76

2 Potential theory for Dirichlet forms

g10 .x; y/ is the so-called reproducing kernel. We note that 0 < g10 .y; y/ < 1 for any y 2 I . If g10 .y; y/ D E1 .g10 .  ; y/; g10 .  ; y// were non-positive, then g10 .x; y/ D 0; 8x 2 I , and v D 0; 8v 2 F 0 , by (2.1.16), arriving at a contradiction. Let us show that Cap.¹yº/ D 1=g10 .y; y/ . > 0/: .2:1:17/ Take the 1-equilibrium potential pn .2 F 0 / of the open interval Jn D .y  1=n; y C 1=n/. Just as in the proof of Lemma 2.1.1, pn converges to a function p 2 F 0 with metric E1 and, consequently, pointwise. p is characterized by the property p 2 F 0;

p.y/ D 1;

E1 .p; v/  0;

8v 2 F 0 ;

v.y/  0:

.2:1:18/

In fact, p.y/ D 1 follows from the fact that the continuous function pn equals 1 m-a.e. on Jn . If v 2 F 0 is non-negative at y, then vı .x/ D v.x/ C ıg10 .x; y/ is positive on a neighbourhood of y for each ı > 0. Hence , E1 .p; vı / D E1 .p; v/ C ıE1 .p; g10 .  ; y// D limn!1 E1 .pn ; vı /  0. By letting ı # 0, we get the second property in (2.1.18). A comparison of (2.1.16) with (2.1.18) gives p.x/ D g10 .x; y/=g10 .y; y/:

.2:1:19/

(2.1.17) follows from this and Cap.¹yº/ D E1 .p; p/. Thus every non-empty set has a positive capacity in the present example. In particular, (2.1.18) characterizes the 1-equilibrium potential p of the one point set ¹yº. Using this again we can readily derive another expression of p: ´ u1 .x/=u1 .y/ if x  y; p.x/ D .2:1:20/ u2 .x/=u2 .y/ if x  y; where u1 .resp. u2 / is a strictly increasing (resp. decreasing) positive 1-harmonic function on I such that u1 .r1 / D 0 .resp. u2 .r2 / D 0/ if r1 .resp. r2 / is a regular boundary (see Example 1.2.2). (2.1.19), (2.1.20), and the symmetry of g10 .x; y/ then lead us to ´ C u1 .x/u2 .y/ if x  y; .2:1:21/ g10 .x; y/ D C u2 .x/u1 .y/ if x  y; for some constant C > 0. This expression along with (2.1.16) enables us to obtain Z r2 R1 f .x/ D g10 .x; y/f .y/dy; .2:1:22/ r1

for a non-negative Borel f 2 L2 .I I m/.

2.2

Measures of finite energy integrals

77

Let ¹Fn º be an increasing sequence of closed subsets of I such that Cap.I n Fn / ! 0; n ! 1. Then any compact subinterval of I is contained in some Fn because Cap.¹yº/ is a positive continuous function on I by virtue of (2.1.17) and (2.1.21). In particular, the quasi continuity reduces to the ordinary continuity. Quite similar properties hold in the Dirichlet forms on “finitely ramified” fractal sets in Rd .4 Exercise 2.1.1. Check the following: (i) The first five lemmas and two theorems hold under the weaker condition that X is a Hausdorff space with a countable base, m is a -finite Borel measure on X with suppŒm D X and .E; F / is a Dirichlet form on L2 .XI m/. (ii) The capacitability (2.1.6) of Cap holds when additionally X is a Lusin space. (iii) Theorems 2.1.3, 2.1.4 and 2.1.5 remain valid if, in addition to the condition in (i) above, F \ C.X/ is E1 -dense in F .

2.2

Measures of finite energy integrals

A positive Radon measure  on X is said to be of finite energy integral if Z p jv.x/j.dx/  C E1 .v; v/; v 2 F \ C0 .X/; .2:2:1/ X

for some positive constant C . A positive Radon measure  on X is of finite energy integral if and only if there exists for each ˛ > 0 a unique function U˛  2 F such that Z E˛ .U˛ ; v/ D v.x/.dx/; 8v 2 F \ C0 .X/: .2:2:2/ X

We call U˛  an ˛-potential. Let ¹T t ; t > 0º be the Markovian semi-group on L2 .XI m/ associated with the Dirichlet form E: u 2 L2 .XI m/ is called ˛-excessive (with respect to ¹T t ; t > 0º) if u  0; e ˛t T t u  u m-a.e.; 8t > 0: .2:2:3/ Theorem 2.2.1. The following conditions are equivalent to each other for u 2 F and ˛ > 0. (i) u is an ˛-potential. (ii) u is ˛-excessive. 4 J.

Kigami [1], R. L. Dobrushin and S. Kusuoka [1], M. Fukushima [20].

78

2 Potential theory for Dirichlet forms

(iii) u  0; ˇGˇ C˛ u  u m-a.e., 8ˇ > 0, where Gˇ is the resolvent of T t . (iv) E˛ .u; v/  0; 8v 2 F ; v  0 m-a.e. (v) E˛ .u; v/  0; 8v 2 F \ C0 .X/; v  0. Proof. The implication (ii) ) (iii) is clear from (1.3.1). By (1.3.18) and the resolvent equation, we have E˛ .u; v/ D limˇ !1 ˇ.u  ˇGˇ C˛ u; v/ which gives the implication (iii) ) (iv). Suppose that (iv) is satisfied. Then we can see by using (2.1.5) that u is the unique element in the convex set Lu D ¹w 2 F I w  u m-a.e.º minimizing E˛ .w; w/. Since juj 2 Lu ; u D juj  0. Moreover, .u  e ˛t T t u; v/ D .u; v  e ˛t T t v/ D E˛ .u; G˛ v  e ˛t T t G˛ v/, which is for any non-negative v 2 L2 .XI m/ because G˛ v  e ˛t T t G˛ v D Rnon-negative t ˛s Ts vds is then non-negative as well. Hence u is ˛-excessive. 0 e (i) ) (v) and (iv) ) (v) are trivial. Suppose that (v) is satisfied. Choose for any non-negative v 2 F a sequence vn 2 F \ C0 .X/; E1 -convergent to v. According to Theorem 1.4.2, vnC is then E1 -convergent to v as well and E˛ .u; v/ D limn!1 E˛ .u; vnC /  0, which proves (iv). Assume (v) and let I.v/ D E˛ .u; v/; v 2 F \ C0 .X/. Consider any compact set K and choose non-negative vK 2 F \ C0 .X/ such that vK  1 on K. Then jI.v/j  kvk1  I.vK / for any v 2 F \ C0 .X/ with suppŒv  K. Since any function w 2 C0 .X/ can be approximated uniformly on X by functions v 2 F \C0 .X/ with suppŒv  suppŒw by virtue of Lemma 1.4.2, the above inequality assures that I can be extended uniquely to a positive linear functional on C0 .X/. Hence u is an ˛-potential of a unique positive Radon measure, and so we obtain (i).

Corollary 2.2.1. If u1 ; u2 2 F are ˛-potentials, then so are u1 ^ u2 and u1 ^ 1. This assertion is indeed trivial for ˛-excessive functions. It is sometimes convenient to replace the space F \ C0 .X/ in the preceding development by a certain core of E. Lemma 2.2.1. Let C be a special standard core of E. (i) Each equivalent condition of Theorem 2.2.1 is also equivalent to E˛ .u; v/  0;

8v 2 C;

v  0:

.2:2:4/

(ii) A positive Radon measure  on X is of finite energy integral if and only if inequality .2:2:1/ holds for any v 2 C.

2.2

Measures of finite energy integrals

79

Proof. (i) It suffices to derive the condition (v) of Theorem 2.2.1 by assuming (2.2.4). Take any non-negative v 2 F \ C0 .X/. By virtue of Theorem 1.4.2, we may assume that v  1. For any " > 0, choose a function " appearing in the definition of the standard core C. We take also a non-negative function wK 2 C such that wK D 1 on K D suppŒv. Choose vn 2 C which is E1 -convergent to v and let wn D wK  " .vn /. The wn are members of C with uniformly bounded E1 -norm by Theorem 1.4.2. Since wn is L2 -convergent to v, it converges E˛ weakly to v and E˛ .u; v/ D limn!1 E˛ .u; wn /  "  E˛ .u; wK /; " > 0 being arbitrary, we get (v) of Theorem 2.2.1. (ii) Suppose that  satisfies (2.2.1) Rfor any v 2 C. Since C is E˛ -dense in F , there exists u 2 F with E˛ .u; v/ D X vd; 8v 2 C. In particular, u satisfies (2.2.4) and consequentlyR (i) impliesRthat u D U˛  for some . It is enough to prove that  D . But X vd D X vd; 8v 2 C, and, for any v 2 C0 .X/, there is a sequence vn 2 C uniformly convergent to v. Let K D suppŒv. Then wK  vn 2 C has a common compact support and converges uniformly to v again. Therefore  D . Exercise 2.2.1. Let E be the Dirichlet form . 12 D; H 1 .Rd // on L2 .Rd /; d  2. Let  be the Lebesgue measure on a .d  1/-dimensional hyperplane F  Rd and B be a bounded subset of F . Show that 1B   is of finite energy integral with respect to E. We denote by S0 the family of all positive Radon measures of finite energy integrals. We next show that each  2 S0 can be approximated by certain measures n 2 S0 which are absolutely continuous with respect to m. Lemma 2.2.2. For  2 S0 and ˛ > 0, let gn D n.U˛   nGnC˛ .U˛ //;

n D 1; 2; : : : :

.2:2:5/

Then gn  m converges vaguely to  and G˛ gn converges E˛ -weakly to U˛ . Proof. Applying Theorem 2.2.1 to u D U˛ , we see that gn  0 m-a.e. Moreover, E˛ .G˛ gn ; v/ D .gn ; v/ D n.unG nC˛ u; v/ ! E˛ .u; v/; n ! 1; v 2 F . R In particular, limn!1 .gn ; v/ D X v.x/.dx/; 8v 2 F \ C0 .X/. Lemma 2.2.3. Each measure in S0 charges no set of zero capacity. Proof. It suffices to prove for  2 S0 p p .G/  E1 .U1 ; U1 /  Cap.G/;

8G 2 O0 :

.2:2:6/

80

2 Potential theory for Dirichlet forms

This follows from the preceding lemma with ˛ D 1: Z gn .x/m.dx/  lim .gn ; eG / .G/  lim n!1

n!1 G

p p D E1 .U1 ; eG /  E1 .U1 ; U1 / E1 .eG ; eG /; which equals the right-hand side of (2.2.6) in view of (2.1.4). Theorem 2.2.2. For any  2 S0 , Z E˛ .U˛ ; v/ D

e  L1 .XI / F

X

e v .x/.dx/;

(2.2.7) ˛ > 0; v 2 F :

(2.2.8)

Here e v denotes any quasi continuous modification (in the restricted sense) of v 2F. Proof. For any v 2 F , we choose a sequence vn 2 F \ C0 .X/ which is E1 -convergent to v. By Theorem 2.1.4 and Lemma 2.1.4, a subsequence nk exists v; e v being any fixed quasi continuous modsuch that vnk converges q.e. on X to e ification of v. Since the inequality (2.2.1) holds for every vn , we have from the preceding lemma and Fatou’s lemma that Z Z je v .x/  vn .x/j.dx/ D lim jvnk .x/  vn .x/j.dx/ X

X nk !1

 C  lim

nk !1

q

E1 .vnk  vn ; vnk  vn /;

v . By letting which implies that e v 2 L1 .XI / and vn is L1 .XI /-convergent to e n tend to infinity in equation (2.2.2) with v being replaced by vn , we can arrive at (2.2.8). For  2 S0 we set

E˛ ./ D E˛ .U˛ ; U˛ /:

.2:2:9/

RWe may now call this the ˛-energy integral of  because we have E˛ ./ D X U˛ .x/.dx/ from Theorem 2.2.2. We now present several other consequences of Theorem 2.2.2. We shall first formulate the following maximum principle.

e

Lemma 2.2.4. Consider the two ˛-potentials u1 D U˛  and u2 D U˛ , ˛ > 0; ;  2 S0 .

2.2

Measures of finite energy integrals

81

u2 -a.e., then u1  u2 m-a.e. (i) If e u1  e (ii) If e u1  C -a.e. for some constant C , then u1  C m-a.e. Proof. (i) Set u D u1 ^ u2 . Then by (2.2.8) Z Z E˛ .u1 ; u/ D e u.x/.dx/ D e u1 .x/.dx/ D E˛ .u1 ; u1 /: X

X

Since u is also an ˛-potential and u  u1 m-a.e., E˛ .uu1 ; uu1 / D E˛ .u; u u1 /  E˛ .u1 ; u  u1 /  0, proving u1 D u  u2 m-a.e. (ii) It suffices to set u2 D u1 ^ C . Let us introduce a subset S00 of S0 defined by S00 D ¹ 2 S0 W .X/ < 1; kU1 k1 < 1º;

.2:2:10/

where k  k1 denotes the norm in L1 .XI m/. Lemma 2.2.4 leads us to Lemma 2.2.5. For any  2 S0 , there exists an increasing sequence ¹Fn º of compact sets such that 1Fn   2 S00 ; Cap.K n Fn / ! 0;

n D 1; 2; : : : ;

n ! 1;

for any compact set K:

e

Proof. For  2 S0 , take a quasi continuous modification U1  of U1  and consider an associated nest ¹Fn0 º. Let ¹En º be an increasing sequence of relatively compact open sets such that E n  EnC1 ; En " X. If we let

e

Fn D ¹x 2 Fn0 \ E n I U1 .x/  nº;

n D 1; 2; : : : ;

then ¹Fn º is an increasing sequence of compact sets and, for any compact K,

e

lim Cap.K n Fn /  lim ŒCap.X n Fn0 / C Cap.¹U1  > nº/;

n!1

n!1

e

which vanishes by Lemma 2.1.6. f1 .1Fn  /  U1   n q.e. on Fn ; U1 .1Fn  /  n m-a.e. by virtue of Since U Lemma 2.2.4. S We note that  2 S0 vanishes on X n 1 nD0 Fn for the sets Fn of Lemma 2.2.5 because of Lemma 2.2.3. Lemma 2.2.6. The following conditions are equivalent for u 2 F and a closed set F .

82

2 Potential theory for Dirichlet forms

(i) u D U˛ ;  2 S0 , with suppŒ  F . v  0 q.e. on F . (ii) E˛ .u; v/  0; 8v 2 F ; e (iii) E˛ .u; v/  0; 8v 2 F \ C0 .X/; v  0 on F . Proof. (i) ) (ii) is an immediate consequence of Theorem 2.2.2. (ii) ) (iii) is trivial. (iii) ) (i) is clear from Theorem 2.2.1 and Lemma 1.4.2. In the previous section, we have defined for any set B  X with Cap.B/ < 1 the 1-equilibrium potential eB characterized as a unique element of F satisfying e eB D 1 E1 .eB ; v/  0;

q.e. on B;

(2.2.11)

8v 2 F ; e v  0 q.e. on B:

(2.2.12)

In view of Lemma 2.2.6, eB D U1 B with a unique B 2 S0 . We call B the 1-equilibrium measure of B: suppŒB  is contained in B. In particular we have from Theorem 2.1.5 and Theorem 2.2.2 that, for any compact set K, Cap.K/ D E1 .eK ; eK / D K .K/:

.2:2:13/

Exercise 2.2.2. Show that the capacity of a compact set K can be computed as

e

Cap.K/ D sup¹.K/ W  2 S00 ; suppŒ  K; U1   1 q.e.º: Exercise 2.2.3 (The Gauss quadratic form). Show that the 1-equilibrium measure K of a compact set K is the unique element in the class ¹ 2 S0 W suppŒ  Kº minimizing the quadratic form G./ D E1 ./  2.K/: Theorem 2.2.3. The following conditions are equivalent for a Borel set B  X: (i) Cap.B/ D 0. (ii) .B/ D 0; 8 2 S0 . (iii) .B/ D 0; 8 2 S00 . Proof. (i) ) (ii) and (iii) ) (ii) are contained in Lemma 2.2.3 and Lemma 2.2.5 respectively. If a Borel set B has a positive capacity, then Cap.K/ > 0 for some compact set K  B by (2.1.6). Hence K .B/ D K .K/ > 0 by (2.2.13); proving (ii) ) (i). The second assertion of the next lemma is useful in computing the capacity.

2.2

Measures of finite energy integrals

83

Lemma 2.2.7. Let C be a special standard core of E. (i) Each condition of Lemma 2:2:6 is also equivalent to the following weaker one: E˛ .u; v/  0; 8v 2 C; v  0 on F: .2:2:14/ (ii) For any compact set K Cap.K/ D inf E1 .u; u/ u2C K

.2:2:15/

where C K D ¹u 2 C W u.x/  1; 8x 2 Kº. Proof. (i) On account of Lemma 2.2.1, we know that (2.2.14) implies u D U˛  for some  2 S0 . For any v 2 C0 .X/ with suppŒv D K  X n F; C admits a function w with w D 1 on K and w D 0 on F . ChooseRvn 2 C uniformly Rconvergent to v. Then w vn 2 C converges uniformly to v and vd D limn!1 w vn d, which vanishes by (2.2.14). Hence suppŒ  F . (ii) Consider a sequence un 2 C K minimizing the E1 -norm. By virtue of (2.1.5), un is E1 -convergent to some u0 2 F . In view of (2.1.12), it is sufficient to prove that u0 D eK by checking the conditions (2.2.11) and (2.2.12) for u0 and B D K. For any v 2 C with v  0 on K; E1 .un C "v; un C "v/  E1 .u0 ; u0 /; 8" > 0, and we see that u0 satisfies (2.2.14) by letting n ! 1 and then " # 0. By the first assertion (i), we can conclude that u0 satisfies (2.2.12). (2.2.11) can be obtained by noting that 1=n .un / is also a minimizing sequence. We now turn to the study of a class S of measures larger than S0 which will play a primary role in Chapter 5. Let us call a (positive) Borel measure  on X smooth if it satisfies the following conditions: (S.1)  charges no set of zero capacity, (S.2) there exists an increasing sequence ¹Fn º of closed sets such that .Fn / < 1;

n D 1; 2; : : : ;

lim Cap.K n Fn / D 0 for any compact set K:

n!1

(2.2.16) (2.2.17)

Let us note that  then satisfies 1   [ Fn D 0:  Xn

.2:2:18/

nD1

An increasing sequence ¹Fk º of closed sets satisfying condition (2.2.17) will be called a generalized nest to distinguish it from the nest introduced in the preceding

84

2 Potential theory for Dirichlet forms

section. If further each Fn is compact, we call it a generalized compact nest. Given a smooth measure , we call ¹Fn º satisfying (2.2.16) and (2.2.17) a generalized nest associated with . We denote by S the family of all smooth measures. The class S is quite large and it contains all positive Radon measures on X charging no set of zero capacity. In particular, S S0 in view of Lemma 2.2.3. We claim, however, that any measure in S can be approximated by measures in S0 and in S00 as well. First, we need two lemmas. Lemma 2.2.8. Let  be a bounded positive Borel measure on X. If .A/  C  Cap.A/ for any Borel set A and for a positive constant C , then  2 S0 . Proof. For any non-negative v 2 F \ C0 .X/ with E1 .v; v/ D 1, we have by using (2.1.10) Z 1 X v.x/.dx/  .X/ C 2kC1 .¹x W 2k  v.x/ < 2kC1 º/ X

kD0

 .X/ C C

1 X

2kC1 Cap.¹x W v.x/  2k º/

kD0

 .X/ C C

1 X

2kC1 22k D .X/ C 4C;

kD0

which shows that  2 S0 . Lemma 2.2.9. Let  be a bounded positive Borel measure on X charging no set of zero capacity. Then there exists a decreasing sequence ¹Gn º of open sets such that Cap.Gn / ! 0; n

.Gn / ! 0;

n ! 1;

.A/  2 Cap.A/ for any Borel set A  X n Gn :

(2.2.19) (2.2.20)

Proof. Fix n and let ˛ D inf¹2n Cap.A/.A/I A is a Borel subset of Xº. Clearly .X/  ˛. If ˛ < 0, choose an open set B1 such that 2n Cap.B1 /  .B1 /  ˛=2. Then ˛1 D inf¹2n Cap.A/  .A/I A  X n B1 º is not less than ˛=2 because ˛  2n Cap.A[B1 /.A[B1 /  ¹2n Cap.A/.A/ºC¹2n Cap.B1 /.B1 /º. If ˛1 < 0, choose a relatively open set B2  X nB1 such that 2n Cap.B2 /.B2 /  ˛1 =2. Continuing in this fashion we find open sets ¹B1 [ B2 [    [ Bk º such that ´ 2n Cap.B1 [    [ Bk /  .B1 [    [ Bk /  0 .2:2:21/ 2n Cap.A/  .A/  2k ˛; 8A  X n .B1 [    [ Bk /:

2.2

Measures of finite energy integrals

85

S n Let us put Gn0 D 1 kD1 Bk . Then by (2.2.21) 2 Cap.A/  .A/ for any Borel 0 n 0 Furthermore, 2 Cap.Gn /  .Gn0 /  .X/ < 1. Then it subset A of X n Gn . S 0 suffices to set Gn D 1 mDn Gm , because ¹Gn º is a decreasing sequence of open nC1 .X/ ! 0; n ! 1. Since  charges no set of zero sets with Cap.GT n/  2 capacity, 0 D . n Gn / D limn!1 .Gn /. Theorem 2.2.4. The following conditions are equivalent for a positive Borel measure  on X. (i)  2 S . (ii) There exists a generalized nest ¹Fn º satisfying .2:2:18/ and 1Fn   2 S0 for each n. (iii) There exists a generalized compact nest ¹Fn º satisfying .2:2:18/ and 1Fn   2 S00 for each n. Proof. “(iii) ) (i)” is clear. The implication (i) ) (ii) follows from the preceding two lemmas. In fact, when  is a bounded positive Borel measure charging no set of zero capacity, it is sufficient to adopt as Fn the complementary set of Gn of Lemma 2.2.9. For a general measure  2 S, let ¹El º be an associated generalized nest. Since .l/ l D 1El   is bounded for each l; l admits a sequence ¹Fn º with the properSn .l/ ties in the statement of Theorem 2.2.4 (ii). We set Fn D lD1 ¹El \ Fn º; n D 1; 2; : : : : The ¹Fn º are then increasing closed sets such that 1Fn   2 S0 for each n. (2.2.16) and (2.2.17) for ¹Fn º follow from the inclusion K n Fn  .K n El / [ .l/ .K n Fn / for any compact set K. We can prove the implication (ii) ) (iii) by using Lemma 2.2.5 in a manner as in the preceding paragraph. In the rest of this section, we suppose that .E; F / is transient and let .Fe ; E/ be the extended Dirichlet space. We say that a positive Radon measure  on X is .0/ of finite (0-order) energy integral ( 2 S0 in notation) if the inequality (2.2.1) holds with E1 on the right-hand side being replaced by E. Then the equation (2.2.2) with ˛ D 0 determines a unique function U 2 Fe . This is called the (0-order) potential of . Evidently .0/ S0  S0 .2:2:22/ .0/

and in particular any  2 S0 charges no set of zero capacity by Lemma 2.2.3. Recalling Theorem 2.1.6, we can prove in exactly the same manner as in the proof of Theorem 2.2.2 its 0-order counterpart:

86

2 Potential theory for Dirichlet forms

e e  L1 .XI / and Theorem 2.2.5. For any  2 S0 ; F Z e v .x/.dx/; v 2 Fe : E.U; v/ D .0/

.2:2:23/

X

e e denotes the set of all quasi continuous functions in Fe . Here F Particularly we have the notion of the (0-order) energy integral Z f U.x/.dx/ E./ D E.U; U/ D X

.0/

of  2 S0 . By this theorem, we get a 0-order version of a part of Theorem 2.2.1: Lemma 2.2.10. The next conditions are equivalent for u 2 Fe . (i) u is a potential. (ii) E.u; v/  0; 8v 2 Fe ; v  0 m-a.e. (iii) E.u; v/  0; 8v 2 F \ C0 .X/; v  0. The implication (iii) ) (i) can be proved in the same way as in the proof of Theorem 2.2.1. Condition (ii) implies that u is non-negative. In order to obtain a 0-order version of Corollary 2.2.1, we prove the next lemma of the approximation: .0/

Lemma 2.2.11. For  2 S0 ; U˛  is E-convergent to U as ˛ # 0. Further lim ˛.U˛ ; v/ D 0;

˛#0

8v 2 F \ C0 .X/:

Proof. Using (2.2.2) and (2.2.23), we have

e

e f i  hUe; i;  hU;

kU˛   Uˇ k2"  hU˛ ; i  hUˇ ; i; kU  U˛ k2"

˛

0 0 and for any subset E of Rd by C˛;2 .E/ D inf¹kf k22 W f 2 L2C .Rd /; G˛  f .x/  1; x 2 Eº;

.2:4:13/

106

2 Potential theory for Dirichlet forms

which is a Choquet capacity on Rd and an outer capacity as well in view of Theorem 2.3.11 and Proposition 2.3.5 of [AH]. By virtue of Proposition 2.3.13 of [AH], we have for any compact set K  Rd 2 C˛;2 .K/ D inf¹kukL ˛;2 .Rd / W u 2 ; u.x/  1; x 2 Kº:

This identity combined with (2.4.12) and the inequality in (1.4.31) leads us to the relation C˛=2;2 .E/  Cap.E/  21˛=2 C˛=2;2 .E/ .2:4:14/ holding for any compact set E; which in turn extends to an arbitrary set E: When ˛ < d; E is transient and its extended Dirichlet space .Fe ; E/ is exactly the same as the Riesz potential space of index ˛2 in view of (1.5.19). Analogously to (2.4.13), the Riesz capacity on Rd of index ˛ is defined for 0 < ˛ < d2 and for any subset E of Rd by P ˛;2 .E/ D inf¹kf k22 W f 2 L2C .Rd /; I˛  f .x/  1; x 2 Eº: C

.2:4:15/

In particular, CP ˛;2 .E/  C˛;2 .E/ for any set E because I˛  G˛ : We show that Cap.0/ .E/ D CP ˛=2;2 .E/ for any E  Rd :

.2:4:16/

By the 0-order counterpart of Theorem 2.1.5 (i), we have for any E  Rd Cap.0/ .E/ D inf¹E.u; u/ W u 2 Fe ; u.x/ Q  1 8x 2 E n N for some N with Cap.0/ .N / D 0º;

.2:4:17/

where uQ denotes a Cap.0/ -quasi continuous version of u: Since I˛=2  f is easily verified to be Cap.0/ -quasi continuous, we see from (2.4.15) and (2.4.17) that the inequality  holds in (2.4.16). On the other hand, by Remark after Theorem 2.3.10 of [AH], CP ˛;2 .E/ D inf¹kf k22 W f 2 L2 .Rd /; I˛  f .x/  1; 8x 2 E n N for some N with CP ˛;2 .N / D 0º:

.2:4:18/

If Cap.0/ .N / D 0; then C˛=2;2 .N / D 0 by Theorem 2.1.6 and (2.4.14) , and consequently CP ˛=2;2 .N / D 0: Therefore we get the inequality  in (2.4.16) from (2.4.17) and (2.4.18). We let B.x; r/ D ¹y 2 Rd W jy  xj < rº; x 2 Rd ; r > 0:

2.4

Capacities and Sobolev type inequalities

107

Theorem 2.4.3. Let 0 < ˛  2 and q > 2: Consider a Borel measure  on Rd : (i) Assume that 0 < ˛ < d: Then the Sobolev type inequality 2 kukL q .Rd I /  S E.u; u/;

u 2 F \ C0 .Rd /;

.2:4:19/

holds for some positive constant S if and only if sup

r d C˛ .B.x; r//2=q < 1:

.2:4:20/

x2Rd ; r>0

(ii) Assume that 0 < ˛ < d: Then the Sobolev type inequality 2 kukL q .Rd I /  S E1 .u; u/;

u 2 F \ C0 .Rd /;

.2:4:21/

holds for some positive constant S if and only if sup

r d C˛ .B.x; r//2=q < 1:

.2:4:22/

x2Rd ; 0 0, let  .t/ be a real function as given in the MarkovDŒE. ian condition (E.4) in 1:1 and put Q  .t/ D .1=˛/˛ .˛t / and wn D Q  .vn /. In the same way as in the proof of the implication (c))(b) of Theorem 1.4.1, we have .wn /  .vn /. Therefore, limn!1 .wn / D .G˛ u/ and wn is EN 1 -convergent to G˛ u. In particular, a subsequence of wn converges to G˛ u m-a.e. Since   wn  1=˛ C  m-a.e.,   G˛ u  1=˛ C  m-a.e., for arbitrary , we get the Markovian nature of ˛G˛ . Theorem 3.1.2. Assume that a closable Markovian symmetric form E on L2 .XI m/ satisfies the following conditions (i) and (ii): (i) DŒE is a dense subalgebra of C0 .X/, (ii) there exists for any compact set K and relatively compact open set G K a non-negative function u 2 DŒE such that u.x/ D 1 for x 2 K and u.x/ D 0 for x 2 X n G. N FurIf E has the local property, then so does its smallest closed extension E. N thermore, E is a regular Dirichlet form possessing DŒE as its special standard core. Proof. The last statement is clear from Theorem 3.1.1 and condition (i). Assuming N the local property of E, we claim the same property of E: N N have disjoint compact supports: E.u; v/ D 0 if u; v 2 DŒE

(3.1.1)

We only give the proof of (3.1.1) when v 2 DŒE because the general case that v 2 N can be proved in a similar way. We may further assume 0  u  1 m-a.e. In DŒE N fact, by putting uC D .0 _ u/ ^ `; u D ..`/ _ u/ ^ 0 for a general u 2 DŒE, ` ` C C  N we have suppŒu`   suppŒu, suppŒu`   suppŒu, and lim`!1 ¹E.u` ; v/ C N  ; v/º D E.u; N E.u v/ by Theorem 1.4.2. ` Let us prove (3.1.1) under the above assumptions. Choose un 2 DŒE EN 1 -convergent to u. By virtue of (ii), there exists w 2 DŒE such that w.x/ D 1; 8x 2 suppŒu and suppŒw \ suppŒv D ;. Put uQ n D 1=n .un /  w with 1=n being the function in the Markovian condition (E.4). Then uQ n 2 DŒE 2 is E1 .uQ n ; uQ n /pis uniformly bounded because p p L -convergent to u. Furthermore E1 .uQ n ; uQ n /  .1 C 1=n/ E1 .w; w/ C kwk1 E1 .un ; un / by Theorem 1.4.2. Therefore, uQ n is weakly convergent to u with respect to EN1 ; in particular, EN1 .u; v/ D limn!1 E1 .uQ n ; v/ D 0.

3.1

Closability and the smallest closed extensions

111

Exercise 3.1.1. Under the assumptions of Theorem 3.1.2, suppose that E has the strong local property. Show that EN also possesses the strong local property. Let D be a domain of Rd and m be a positive Radon measure on D with suppŒm D D. In the remainder of this section, we pay special attention to the Markovian symmetric form ´ R @u.x/ @v.x/ Pd E.u; v/ D i;j D1 D @xi @xj ij .dx/ (3.1.2) DŒE D C01 .D/ which appeared in Example 1.2.1 and ask ourselves the question “Under what conditions on m and ij is the form (3.1.2) closable on L2 .DI m/?” When (3.1.2) is closable, then its smallest closed extension gives rise to a regular Dirichlet form on L2 .DI m/ with the local property on account of the preceding two theorems. In the one dimensional case a satisfactory answer to the above question is known. We start however with the multidimensional cases. (1ı ) Absolutely continuous case Consider the case where m is the Lebesgue measure and ij is absolutely continuous, i.e., ij .dx/ D aij .x/dx; 1  i; j  d; (3.1.3) where aij are locally integrable Borel functions on D such that for any  2 Rd and x 2 D d X

aij .x/i j  0;

aij .x/ D aj i .x/;

1  i; j  d:

(3.1.4)

i;j D1

If ¹aij º satisfies one of the following two conditions, then the form (3.1.2) is closable on L2 .D/: .1ı :a/ aij 2 L2loc .D/; @x@ aij 2 L2loc .D/; 1  i; j  d . i Pd ı 2 d .1 :b/ uniform ellipticity W 9ı > 0; i;j D1 aij .x/i j  ıjj ; 8 2 R ; 8x 2 D. Here the derivatives are taken in theP distribution sense. /; D.S / D C01 .D/, Assume .1ı :a/. Then S u.x/ D di;j D1 @x@ .aij .x/ @u.x/ @x i

j

define a symmetric operator S on L2 .D/ and E.u; v/ D .u; S v/; u; v 2 D.S /. This expression implies that E satisfies the condition (1.1.3). Hence E is closable. Assume .1ı :b/ instead of .1ı :a/. Suppose that u` 2 C01 .D/ satisfies lim`;m!1 E.u`  um ; u`  um / D 0 and lim`!1 .u` ; u` / D 0. Then ¹u` º is

112

3 The scope of Dirichlet forms

Cauchy with respect to the Dirichlet integral D. Since D is a special form satisfying condition (1ı :a), we have lim`!1 D.u` ; u` / D 0. Subtracting a subsequence if necessary, we may assume @u` =@xi converges to zero a.e. on D, 1  i  d . Then by Fatou’s lemma  X  d @.u`  um / @.u`  um / aij dx @xi @xj D `!1

Z E.um ; um / D

lim

i;j D1

 lim inf E.u`  um ; u`  um /; `!1

which can be made arbitrarily small for a sufficiently large m. Here we give another closability condition in which aij may satisfy neither .1ı :a/ nor .1ı :b/. Assume D D Rd and assume further that for any 1  i; j  d , .1ı :c/ the distribution derivative .@=@xk /aij belongs to L2loc .Rd / for 1  k  d 1, and the derivative .@=@xd /aij .x/ in the ordinary sense exists only on ¹x 2 Rd W xd ¤ 0º. Then (3.1.2) is closable. To see this, put x 0 D .x1 ; : : : ; xd 1 / for x D .x1 ; : : : ; xd 1 ; xd / and set F 0 D ¹u 2 C01 .Rd / W 9ı > 0; u.x/ depends only on x 0 when xd 2 .ı; ı/º: In the same way as in the proof .1ı :a/, we can see that the form (3.1.2) with the domain DŒE being restricted to F 0 is closable on L2 .Rd /. Next define for any u 2 C01 .Rd / 8 0 ˆ 2ı  xd  2ı 2ı ˆ : u.x 0 ; xd C 2ı/ xd < 2ı and put uı .x/ D jı  vı .x/ with the mollifier jı of Exercise 1.2.1. Obviously uı 2 F 0 and suppŒuı  is contained in a compact set independent of ı > 0. When ın # 0; uın .x/ ! u.x/ and .@=@xk /uın .x/ ! .@=@xk /u.x/ boundedly on the set ¹x 2 Rd W xd ¤ 0º. Hence limn!1 E1 .uın  u; uın  u/ D 0, proving the closability of (3.1.2). The following example is a special case of (3.1.2) satisfying .1ı :c/: ´ R P R P @u @v @u @v dx C C2 Rd diD1 @x dx E.u; v/ D C1 Rd diD1 @x @x i i i @xi  C (3.1.5) DŒE D C01 .Rd /; where C1 ; C2  0; RdC D ¹x 2 Rd W xd > 0º and Rd D ¹x 2 Rd W xd < 0º.

3.1

Closability and the smallest closed extensions

113

Next we consider the case that m D 2 dx

and ij .dx/ D ıij 2 dx;

1  i; j  d;

(3.1.6)

where is a non-negative locally square integrable Borel function on D. We then obtain the following sufficient condition for closability of the form (3.1.2). Theorem 3.1.3. Let N  D be a closed set of Lebesgue measure zero. Let 2 @ @ 2 L2loc .D n N / .i D 1; : : : ; d /, where @x are distributional L2loc .DI dx/ and @x i i 2 2 derivatives on D n N . Then the form E is closable on L .DI dx/. Proof. Let

Z E

.i /

.u; v/ D

D

@v @u .x/ .x/ .x/2 dx @xi @xi

.1  i  d /:

If every form E .i / is closable, then so is E. Further the closability of E .i / is equivalent to the closability on L2 .DI 2 dx/ of the operator @x@ defined on C01 .D/. 

Hence it suffices to show that the adjoint operator @x@ i L2 .DI 2 dx/1 . Let  2 C01 .D n N /. Then for any 2 C01 .D/      @. / 2 @ ; D ;  @xi @xi 2 dx dx  @ 1 D ; 2

C @xi @ 1 and 2 @x

C i

operator

@  @xi

@ @xi

i

is defined on a dense set in

 @ 2 ; @xi dx  @ ; @xi 2 dx

2 L2 .DI 2 dx/. Therefore the domain of the adjoint

contains C01 .D n N / which is dense in L2 .DI 2 dx/.

.2ı / Superposition of closable forms In order to treat the cases where the ij are not necessarily absolutely continuous with respect to the Lebesgue measure, we indicate a useful method of superposing closable symmetric forms to get new closable symmetric forms inductively. Assume that D D Rd for simplicity and let E be a symmetric form on L2 .Rd / coercive on the Sobolev space of Example 1.2.3 and specifically expressed as follows: ´ R E.u; v/ D ‚ E  .u ; v  /.d / C c  D.u; v/ (3.1.7) DŒE D C01 .Rd /; 1 Cf.

T. Kato [1].

114

3 The scope of Dirichlet forms

c being a positive constant. Here we are given an auxiliary -finite measure space .‚; B.‚/; /, a collection ¹F W  2 ‚º of .d  1/-dimensional hyperplanes F and, for each  2 ‚, a symmetric form E  on L2 .F / with DŒE   D C01 .F /; L2 .F / being based on the (d  1)-dimensional Lebesgue measure m on F . We assume that E  .u ; u / and m .K  / are -integrable in  2 ‚ for every u 2 C01 .Rd / and every compact set K  Rd , where u is the restriction to F of u and K  D K \ F . Theorem 3.1.4. (i) Assume that E  is closable on L2 .F / for each  2 ‚. Then so is the form .3:1:7/ on L2 .Rd /. (ii) If each E  is expressed as .3:1:2/ on the hyperplane F , then the form .3:1:7/ is a Markovian symmetric form possessing the local property. (ii) is clear. The first assertion is a consequence of the following general criterion for closability. Theorem 3.1.5. Suppose that a space X and measures m, m O on it satisfy condition .1:1:7/. Let E and EO be Markovian symmetric forms closable on L2 .XI m/ and O respectively. We assume that DŒE is a dense subset of C0 .X/, L2 .XI m/ O  DŒE; DŒE

O E.u; u/  E.u; u/

O for u 2 DŒE;

(3.1.8)

and that the measure m O belongs to the class S of smooth measures ( 2:2) related to the regular Dirichlet form EN on L2 .XI m/. Then EO is closable on L2 .XI m/. O `  um ; u `  u m / D 0 O satisfies lim`;m!1 E.u Proof. Suppose that ¹u` º  DŒE and lim`!1 .u` ; u` /m D 0. Then, by the assumption (3.1.8) ¹u` º constitutes also a Cauchy sequence with respect to E and hence lim`!1 E.u` ; u` / D 0 by the closability of E. On the other hand, by Theorem 2.2.4, there exists a generalized nest ¹Fn º satisfying 1   [ Fn D 0 and 1Fn  m O 2 S0 for each n: m O Xn nD1

Hence we have

Z X

ju` j1Fn d m O C

p E1 .u` ; u` / ! 0

O Acand, by subtracting a subsequence if necessary, u` converges to zero m-a.e. cording to Theorem 1.5.2 (i), we can conclude that O ` ; u` / D E.0; O 0/ D 0; lim E.u

`!1

which implies the closability of EO on L2 .XI m/.

3.1

Closability and the smallest closed extensions

115

Proof of Theorem 3:1:4. (i) Let Z m. O  / D dx C

‚

m .  /.d /:

O In fact, if ¹u` º  Then the form E defined by (3.1.7) is closable on L2 .Rd I m/. C01 .Rd / satisfies lim E.u`  um ; u`  um / D 0 `;m!1

and lim .u` ; u` /mO D 0;

`!1

then we can choose a subsequence kj such that limj !1 uk D 0 on L2 .F / for j q P -a.e.  2 ‚ and j1D1 E.ukj  ukj C1 ; ukj  ukj C1 / < 1. This means that 1 q X j D1

E  .uk  uk j

j C1

; uk  uk j

j C1

/ 0; jKj D 0; j  j being the Lebesgue measure. Without loss of generality we assume K  .0; 1/. Take a sequence n0 2 C01 .R1 / such that suppŒ n0   .0; 1/ and n0 decreases to the indicator R x function 1K as n " 1. We 0 0 set n .x/ D n .x/  n .x  1/ and n .x/ D 1 n .x/dx. R2 Then n is a C 1 function with suppŒn   .0; 2/ and n .x/2  2 0 n .x/2 dx, boundedly to .x/ D 1K .x/  1K .x  1/, 0  x  R2. Since n converges R2 2 .n ; n / D 0 n .x/2 dx  4 0 n .x/2 dx ! 0, n ! 1, whereas E.n ; n / D R2 R2 2 .x/2 .dx/. Obviously ¹n º n .x/ .dx/ converges to a non-zero limit 0 0 constitutes an E-Cauchy sequence. Hence E is not closable. Necessity of (ii). Suppose that E is closable even though condition (ii) is not satisfied. Then there exists a constant ˛ > 0 such that AQ D S.a/ \ ¹x W a.x/  ˛º is of positive Lebesgue measure. Without loss of generality we assume that the set A D AQ \ .0; 1/ has positive Lebesgue measure. Put 

B D .0; 1/ n A;

Ik;n

k kC1 D ; ; n n

n D 1; 2; : : : ; 0  k  n  1:

Let us define a sequence ¹n º  L1 .R1 / as follows: n D 0 on R1 n .0; 1/;

n D 1 on A

(3.1.12)

if jA \ Ik;n j D 0:

(3.1.13)

and n D 0 on B \ Ik;n

118

3 The scope of Dirichlet forms

If jA \ Ik;n j > 0, then Z Z 1 a dx D 1D Z  and

R B\Ik;n

Ik;n

B\Ik;n

a

1

B\Ik;n

Z dx C

a1 dx

A\Ik;n

1 jAj; ˛

a1 dx C

a1 dx D C1. Hence we can find a constant M > 0 such that Z M ^ a1 dx  1: (3.1.14) B\Ik;n

Accordingly we may set n D ˇ.M ^ a1 / on B \ Ik;n

if jA \ Ik;n j > 0;

(3.1.15)

where ˇ < 0 is chosen so that Z jA \ Ik;n j C ˇ

B\Ik;n

Then the sequence ¹n º satisfies Z n dx D 0 Ik;n

M ^ a1 dx D 0:

(3.1.16)

for 0  k  n  1;

(3.1.17)

on account of (3.1.13), (3.1.15) and (3.1.16). Furthermore Z 1 n2 adx  2 for each k; n B\Ik;n

(3.1.18)

because, by (3.1.14), (3.1.15) and (3.1.16), Z Z  2 2 2 n adx D ˇ M ^ a1 adx B\Ik;n

B\Ik;n

Z

2

 .ˇ/

B\Ik;n

M ^ a1 dx

D ˇjA \ Ik;n j  jA \ Ik;n j2  Denote by EN the closure of E and set Z x D n dx; n 0

x 2 R1 :

1 : n2

3.1

Closability and the smallest closed extensions

119

N To see this, put 'ı D jı  n with Then each n belongs to the domain of E. the mollifier jı of Exercise 1.2.1. Then 'ı belongs to C01 .R1 / and suppŒ'ı  is contained in a compact set independent of ı > 0. When ı` # 0 as ` ! 1, 'ı` converges to n uniformly and the derivative .'ı` /0 D jı`  n is uniformly bounded and convergent to n dx-a.e. Moreover ¹ n º has the following properties: converges to zero in L2 .R1 I m/, N sequence, (b) ¹ n º is an E-Cauchy R N n ; n / D adx  ˛jAj > 0. (c) limn!1 E. A (a) ¹

nº

Indeed, it holds by the definition of n that for x 2 Ik;n Z j n .x/j  jn jdx Ik;n

Z D

Ik;n

Z 1A n dx 

Ik;n

Z

D2

1A n dx Ik;n

Z D2

1R1 nA n dx

Ik;n

1A dx 

2 ; n

and so (a) follows. The property (b) holds since Z N .n  m /2 adx E. n  m ; n  m / D R1 1

Z D

0

.n  m /2 1B adx

Z

2 

1 0

Z n2 1B adx

1 1 2 C n m

C



0

1

 2 m 1B adx

owing to (3.1.18). Finally the property (c) follows from ˇ2 Z ˇ Z Z 1 ˇd n ˇ 1 2 ˇ ˇ n2 1B adx  : ˇ dx  1A ˇ adx D jn  1A j adx D n 0 But the existence of a sequence with the properties (a)–(c) is contradictory to the closability of E.

120

3.2

3 The scope of Dirichlet forms

Formulae of Beurling–Deny and LeJan

Consider X and m satisfying condition (1.1.7). We first give a general representation theorem on regular Dirichlet forms due to Beurling and Deny.2 We next introduce the notions of energy measures and their local parts which will then be shown to obey a transformation rule due to LeJan.3 As an immediate consequence of the rule, we shall obtain a Beurling–Deny expression of the local part E .c/ of the form E. We shall denote the domain DŒE of E by F . Theorem 3.2.1. Any regular Dirichlet form E on L2 .XI m/ can be expressed for u; v 2 F \ C0 .X/ as follows: Z .c/ .u.x/  u.y//.v.x/  v.y//J.dx; dy/ E.u; v/ D E .u; v/ C X X nd

Z C

u.x/v.x/k.dx/:

(3.2.1)

X

Here E .c/ is a symmetric form with domain DŒE .c/  D F \ C0 .X/ and satisfies the strong local property: E .c/ .u; v/ D 0 for u 2 DŒE .c/  and v 2 I.u/;

(3.2.2)

where I.u/ D ¹v 2 DŒE .c/  W v is constant on a neighbourhood of suppŒuº:

(3.2.3)

J is a symmetric positive Radon measure on the product space X  X off the diagonal d and k is a positive Radon measure on X. Such E .c/ ; J , and k are uniquely determined by E. Furthermore, every normal contraction operates on the form E .c/ . We call J and k in the above theorem the jumping measure and the killing measure4 respectively associated with the regular Dirichlet form E. Proof. (I) Uniqueness. It follows from (3.2.1) that Z E.u; v/ D 2 u.x/v.y/J.dx; dy/ X X

2 Cf.

A. Beurling and J. Deny [2]. Y. LeJan [2]. 4 Cf. Theorem 4.5.2. 3 Cf.

(3.2.4)

3.2

Formulae of Beurling–Deny and LeJan

121

disjoint supports. In view of the regularity of E for any u; v 2 F \ C0 .X/ withP p and Lemma 1.4.2, the functions iD1 ui .x/vi .y/ such that ui ; vi 2 F \ C0 .X/, suppŒui \suppŒvi  D ;; 1  i  p, constitute a dense subalgebra of C0 .XX nd /. Hence a positive Radon measure J on X  X n d satisfying (3.2.4) is uniquely determined by E. Next observe the equality Z E.u; v/ D

X X nd

.u.x/  u.y//.v.x/  v.y//J.dx; dy/

(3.2.5)

Z

C

u.x/k.dx/; X

where v is a function in F \ C0 .X/ identically equal to 1 on a neighbourhood of suppŒu. Since such a v exists for any u 2 F \ C0 .X/ by virtue of Theorem 1.4.2, (3.2.5) uniquely determines a positive Radon measure k. As the measures J and k are unique, the form E .c/ of Theorem 3.2.1 is unique as well. (II) The measure J as a vague limit. We utilize the approximating form E .ˇ / of E defined by (1.3.16) together with the positive Radon measure ˇ on X  X determined by Lemma 1.4.1 and (1.4.7). Then Z E

.ˇ /

.u; v/ D ˇ

X X

u.x/v.y/ˇ .dxdy/ ! E.u; v/;

ˇ!1

(3.2.6)

if u; v 2 F \ C0 .X/ and suppŒu \ suppŒv D ;. This implies that there exists a unique positive Radon measure J on X  X n d satisfying (3.2.4) and ˇ ˇ ! J 2

vaguely on X  X n d as ˇ ! 1:

(3.2.7)

In fact, we can see by (3.2.6) that the family ¹ˇˇ ; ˇ > 0º of measures is uniformly bounded on each compact subset of X  X n d and hence a subsequence ˇn ˇn converges as ˇn ! 1 vaguely on X  X n d to a positive Radon measure J . Since J satisfies (3.2.4) and we have already observed the uniqueness of such J , we get (3.2.7). (III) Construction of k and E .c/ . Fix a metric on X compatible with the given topology. Choose a sequence of relatively compact open sets Gl increasing to X and a sequence of numbers ıl # 0 such that the set l D ¹.x; y/ 2 Gl  Gl W

.x; y/  ıl º is a continuous set with respect to the measure J for every l.

122

3 The scope of Dirichlet forms

In the same way as (1.4.8) we can rewrite the approximating form E .ˇ / as follows: for u 2 F \ C0 .X/ such that suppŒu  Gl , Z 1 .ˇ / .u.x/  u.y//2 ˇ .dxdy/ (3.2.8) E .u; u/ D ˇ 2 Gl Gl Z Cˇ u.x/2 .1  ˇGˇ 1Gl .x//m.dx/: Gl

This particularly implies that the family of measures ˇ.1  ˇGˇ 1Gl .x//  m.dx/; ˇ > 0, is uniformly bounded on each compact subset of Gl . Hence a subsequence ˇn " 1 and positive Radon measures kl on Gl exist and for each l ˇn .1  ˇn Gˇn 1Gl /  m ! kl

vaguely on Gl ; ˇn ! 1:

(3.2.9)

In view of (3.2.8), (3.2.9), (3.2.7), we have for every l such that suppŒu  Gl , Z ˇn E.u; u/ D lim .u.x/  u.y//2 ˇn .dxdy/ (3.2.10) 2 ˇn !1 Gl Gl ; .x;y/ 0º of the Friedrichs extension A0 . Since .G˛0 u; u/ is non-decreasing as ˛ # 0, the following symmetric form J 0 on H in the wide sense is well-defined: J 0 .u; v/ D lim .G˛0 u; v/; ˛#0

DŒJ 0  D ¹u 2 H W lim .G˛0 u; u/ < 1º: (3.3.4) ˛#0

Let S  be the adjoint operator of S. We set N˛ D ¹u 2 D.S  / W .˛  S  /u D 0º;

N˛0 D N˛ \ DŒJ 0 

(3.3.5)

for each ˛ > 0. We may interpret the space N˛0 as the space of ˛-harmonic elements with finite (0-order) energy integrals.

132

3 The scope of Dirichlet forms

Lemma 3.3.2. Let A be a non-negative definite self-adjoint operator on H . (i) A 2 A.S / if and only if S  A. (ii) If A 2 A.S /, the Hilbert space .DŒEA ; EA;˛ / can be orthogonally decomposed as (3.3.6) DŒEA  D DŒEN .S /  ˚ HA;˛ ; ˛ > 0; with HA;˛ D N˛ \DŒEA  D N˛0 \DŒEA . Moreover, the following inequality holds: (3.3.7) EA .u; u/  ˛ 2 J 0 .u; u/; u 2 HA;˛ : Proof. (i) If S  A, then A S  , while S  is the smallest closed extension of S .11 (ii) By virtue of Lemma 3.3.1, DŒEN.S /  is a closed subspace of .DŒEA ; EA;˛ /. Denote by HA;˛ its orthogonal complement, i.e., u 2 HA;˛ if and only if u 2 DŒEA  and EA;˛ .u; v/ D 0; 8v 2 D.S /. Since S  A; EA;˛ .u; v/ can then be written as .u; .S C ˛/v/ in view of Corollary 1.3.1 to Theorem 1.3.1. Hence HA;˛ D N˛ \ DŒEA . It only remains to prove the inequality (3.3.7). The following transform Pˇ;˛ defines a one-to-one correspondence between the spaces N˛ and Nˇ .˛; ˇ > 0/ 1 and that P˛;ˇ D Pˇ;˛ : Pˇ;˛ u D u C .˛  ˇ/Gˇ0 u;

u 2 N˛ :

(3.3.8)

In fact, we see from (i) that A0  S  and .ˇ  S  /Gˇ0 u D u. Hence Pˇ;˛ u 2 D.S  / for u 2 N˛ and .ˇ  S  /Pˇ;˛ u D .ˇ  ˛/u C .˛  ˇ/u D 0, that is, Pˇ;˛ u 2 Nˇ . The resolvent equation immediately leads us to P˛;ˇ Pˇ;˛ u D u; u 2 N˛ . Now fix ˛ > 0 and u 2 HA;˛ and put uˇ D Pˇ;˛ u. Using the orthogonality (3.3.6), we have EA;˛ .u; u/ D EA;˛ .u; uˇ / D EA;ˇ .uˇ ; uˇ / C .˛  ˇ/.u; uˇ /  .˛  ˇ/.u; uˇ / D .˛  ˇ/.u; u/ C .˛  ˇ/2 .u; Gˇ0 u/. Therefore, EA .u; u/  .˛  ˇ/2 .u; Gˇ0 u/  ˇ.u; u/. Letting ˇ tend to 0, we arrive at (3.3.7). Lemma 3.3.2 (ii) suggests that the following expression EK is a candidate for the closed symmetric form associated with the Krein extension AK : DŒEK  D DŒEN .S /  C N˛0 ; 8 N ˆ 0. It is then easy to see by making use of Lemma 3.3.2 (i) that the self-adjoint operator AK associated with EK is an extension of S , namely, AK 2 A.S /. Clearly this AK must be the Krein extension of S in view of Lemma 3.3.2 (ii). When S satisfies an additional condition .S u; u/  .u; u/;

8u 2 D.S /

(3.3.11)

for some constant > 0, the expressions (3.3.9) and (3.3.10) of EK can be simplified as DŒEK  D DŒEN .S /  C N0 ; ´ EN .S / .u; v/; u; v 2 DŒEN.S /  EK .u; v/ D 0; otherwise.

(3.3.12) (3.3.13)

Indeed, under the condition (3.3.11), lim˛#0 .G˛0 u; u/ is finite for any u 2 H and hence we arrive at (3.3.12) and (3.3.13) by letting ˛ (formally) tend to 0 in (3.3.9) and (3.3.10) respectively. Moreover, in this case a direct description of the Krein extension AK is possible: D.AK / D D.S  / C N0 ; ´ S  u; u 2 D.S  / AK u D 0; u 2 N0 :

(3.3.14) (3.3.15)

We shall not give a detailed proof of the above mentioned facts on EK and AK because our main concern is a certain subfamily of A.S / defined below rather than the whole family A.S /. Now let X and m be as in 1.1 and S be a non-negative definite symmetric linear operator on L2 .XI m/ such that the associated symmetric form E.S / is Markovian. Define a subfamily AM .S / of A.S / by AM .S / D ¹A 2 A.S / W the semigroup on L2 .XI m/ generated by A is Markovianº:

(3.3.16)

By virtue of Theorem 1.4.1, A 2 A.S / belongs to AM .S / if and only if EA is a Dirichlet form on L2 .XI m/. We call an element of AM .S / a Markovian selfadjoint extension of S . Theorem 3.1.1 implies that the Friedrichs extension A0 of S is an element of AM .S /; indeed, it is the minimum one of AM .S / with semi-order defined by (3.3.3). Natural questions then arise:

134

3 The scope of Dirichlet forms

(I) Is there the maximum element, say AR , in AM .S /? (II) Does AR equal AK ? We give answers to these questions in the simple case that X is a domain D  Rd ; m is dx for the Lebesgue measure dx and   d 1 X @ @ SD aij .x/ ;

.x/ @xj @xi

D.S / D C01 .D/:

(3.3.17)

i;j D1

Here is a strictly positive C 1 .D/-function and aij ; 1  i; j  n, are C 1 .D/functions such that for any  2 Rd n ¹0º and x 2 D d X

aij .x/i j > 0;

aij .x/ D aj i .x/ ; 1  i; j  d:

(3.3.18)

i;j D1

The operator S becomes a symmetric operator on L2 .D; dx/ and E.S / .u; v/ D

d Z X

aij .x/

i;j D1 D

@u @v .x/ .x/dx; @xi @xj

DŒE.S /  D C01 .D/:

(3.3.19) As we saw in §1.2, E.S / is a Markovian symmetric form. The smallest closed extension of E.S / is denoted by E 0 . Note that L2 .DI dx/  L1loc .DI dx/. We then introduce a symmetric form C E on L2 .DI dx/ by C

E .u; v/ D

C

d Z X

aij .x/

i;j D1 D

°

2

@u @u .x/ .x/dx; @xi @xj

DŒE  D u 2 L .D; dx/ W

(3.3.20)

d Z X i;j D1

± @u @u aij .x/ .x/ .x/dx < 1 @xi @xj D (3.3.21)

@u are taken in the sense of Schwartz distribution. DŒE 0  is where the derivatives @x i then the closure of C01 .D/ in DŒE C . Obviously E C is a Dirichlet form.

Lemma 3.3.3. The space DŒE C  is the completion of DŒE C  \ C 1 .D/ with respect to E1C .

135

3.3 Maximum Markovian extensions

Proof. Take any u 2 DŒE C . Let ¹Dn º1 nD1 be a sequence of relatively compact open sets such that D1  DN 1  D2  DN 2     , and Di " D. Set O1 D D1 ; O2 D D2 ; O3 D D3  DN 1 ; : : : ; On D Dn  DN n2 ; : : : ; and let 1 1 1 ¹'n ºP nD1 be a partition of unity associated with ¹On ºnD1 , namely, 'n 2 C0 .On / 1 1 and nD1 'n D 1. Since u'n belong to the Sobolev space H0 .On /, there exC  1 ist functions n 2 C0 .On / such that E1 .u'n  n ; u'n  n /  2n . Put P1 D iD1 i . Then 2 C 1 .D/ and d Z X i;j D1

@.u  / @.u  / aij .x/ .x/ .x/dx C @xi @xj On 3

1 X

Z On

E1C .u'nCi 

.u 

/2 dx

nCi ; u'nCi



nCi /;

iD1

and consequently E1C .u  ; u 

/  9.

Theorem 3.3.1. Let AR be the self-adjoint operator on L2 .DI dx/ corresponding to .E C ; DŒE C /. Then AR is the maximum element of AM .S /. The proof of this theorem is completed by the following two lemmas. Lemma 3.3.4. AR 2 AM .S /. Proof. Let ¹G˛R ; ˛ > 0º be the resolvent of AR and put u D G˛R f with f 2 L2 .DI dx/. Then, it follows from the definition of E C that for  2 C01 .D/ E˛C .u; /

D

D

d Z X

aij .x/

i;j D1 D

@u @ .x/ .x/dx C ˛.u; / @xi @xj

  1 @ @ u.x/  aij .x/ .x/ .x/dx C ˛.u; /

.x/ @xi @xj D

d Z X i;j D1

D .u; .˛  S //: On the other hand, in view of (1.3.7), E˛C .u; / D .f; /. Hence we obtain .u; .˛  S // D .f; /

for  2 C01 :

(3.3.22)

This equation is in turn equivalent to the statement that u 2 D.S  /; .˛  S  /u D f for S of (3.3.17). Therefore AR  S  and hence S  AR by Lemma 3.3.2 (i), proving that AR 2 AM .S /.

136

3 The scope of Dirichlet forms

Lemma 3.3.5. If A 2 AM .S /, then DŒEA   DŒE C  and EA .u; u/  E C .u; u/; u 2 DŒEA . Proof. The operator S is hypoelliptic, namely, if S u is a C 1 -function in the distribution sense, then u becomes a C 1 -function. Accordingly we have N˛ D ¹u 2 L2 .DI dx/ \ C 1 .D/ W .˛  S /u D 0º:

(3.3.23)

Thus it suffices to show that for a fixed ˛ > 0, EA .u; u/ 

d Z X

aij .x/

i;j D1 D

@u @u .x/ .x/dx; @xi @xj

8u 2 N˛ \ DŒEA : (3.3.24)

In fact, by Lemma 3.3.2 (ii) any u 2 DŒEA  can be decomposed as u D u0 C uh , u0 2 DŒE 0 ; uh 2 N˛ \ DŒEA  and EA;˛ .u; u/ D E˛0 .u0 ; u0 / C EA;˛ .uh ; uh /. Hence, if (3.3.24) is true, then uh 2 DŒE C  and consequently u 2 DŒE C . Moreover, EA;˛ .u;u/  E˛0 .u0 ;u0 /CE˛C .uh ;uh / D E˛C .u;u/ because E˛C .u0 ; uh / D 0. In order to prove (3.3.24), consider the resolvent ¹G˛ ; ˛ > 0º on L2 .DI dx/ of A. Since A  S  by Lemma 3.3.2 (i), the function u D G˛ f; f 2 C01 .D/, satisfies the equation (3.3.22). ˛G˛ being assumed to be Markovian, we can use the hypoellipticity of the operator S to get a Markovian resolvent kernel ¹R˛ .x; E/, ˛ > 0º on D such that G˛ f D R˛ f dx-a.e. on D;

f 2 C01 .D/:

(3.3.25)

Then notice that the equation (3.3.22) is valid for f and u D R˛ f whenever jf j and R˛ jf j are locally integrable. Since in this case .Rˇ jf j; jj/ D .jf j; Rˇ jj/ ! 0;

ˇ ! 1; 8 2 C01 .D/;

lim .ˇRˇ f; / D .f; / C lim .Rˇ f; S/ D .f; /;

ˇ !1

ˇ !1

8 2 C01 .D/:

This implies that the relation lim .ˇ.f  ˇRˇ f /; / D .f; S/ for  2 C01 .D/

ˇ !1

(3.3.26)

is valid when jf j and R˛ jf j are locally integrable. We now proceed to the proof of (3.3.24) with the help of the relation (3.3.26). Take any u 2 N˛ \ DŒEA . By the hypoellipticity of S again, we may assume that u is infinitely differentiable and satisfies the equation .˛  S /u.x/ D 0; x 2 D,

3.3 Maximum Markovian extensions

137

.ˇ /

in the ordinary sense. Writing the approximating form EA .u; u/ of EA .u; u/ in the manner of (1.4.8), we get ´ .ˇ / EA .u; u/ D 12 .fˇ ; 1/ (3.3.27) fˇ .x/ D ˇ 2 Rˇ .u.x/  u/2 .x/ C 2ˇu.x/2 .1  ˇRˇ 1.x//: Since fˇ can be rewritten as fˇ D ˇ.u2  ˇRˇ u2 / C 2ˇu.u  ˇRˇ u/ C ˇu2 .1  ˇRˇ 1/ and u is infinitely differentiable, the relation (3.3.26) leads us to   1 2 1 lim .fˇ ; / D (3.3.28) S u  S u  u;  2 ˇ !1 2 d Z X @u @u aij .x/ .x/ .x/.x/dx;  2 C01 .D/: D @xi @xj D i;j D1

Since fˇ  0; 8ˇ > 0, by (3.3.28), we have, for any  2 C01 .D/ such that 0    1, EA .u; u/ D lim

ˇ !1

.ˇ / EA .u; u/



d Z X i;j D1 D

aij .x/

@u @u .x/ .x/.x/dx: @xi @xj

Letting then  " 1, we arrive at (3.3.24). Corollary 3.3.1. The Friedrichs extension of S is a unique element of AM .S / if and only if DŒE C  is identical with DŒE 0 . In what follows, we identify a non-positive definite self-adjoint operator A with the associated closed symmetric form EA : Under this identification, we introduce a family of non-positive definite self-adjoint operators on L2 .DI dx/ by AS i .S / D ¹.E; F / W .E; F / is a Dirichlet form on L2 .D; dx/ such that F C01 .D/; E.u; u/ D E.S / .u; u/ 8u 2 C01 .D/; and u  ' 2 DŒE 0  8u 2 Fb ; 8' 2 C01 .D/º: We call an element of the right-hand side a Silverstein extension of E.S / . For a probabilistic meaning of AS i .S /, see Theorem A.4.4.12 We easily see that .E C ; DŒE C / belongs to AS i .S / and thus AM .S /  AS i .S /. As an application of the regular representation theory in Appendix A.4, we shall show that AS i .S /  AM .S /. 12 For

a detail explanation, see Z.-Q. Chen and M. Fukushima [1].

138

3 The scope of Dirichlet forms

Q FQ / be a regular representation of .D; dx; Q m; For .E; F / 2 AS i .S /, let .D; Q E; E; F / with respect to a certain subalgebra L of L1 .D; dx/ satisfying condition (L) in A.4. Let ˆ denote the isomorphism between the two Dirichlet spaces. Since C01 .D/  DŒE 0 , we can assume that C0 .D/  L:

(3.3.29)

Q FQ / and ˆ possess following properties. Q m; Then .D; Q E; Lemma 3.3.6. Let u; v; w be functions in C0 .D/ such that suppŒu \ suppŒv D ; and w D k (constant) on a neighbourhood of suppŒu. (i) suppŒˆ.u/ \ suppŒˆ.v/ D ;. (ii) ˆ.w/ D k on a neighbourhood of suppŒˆ.u/. Q (iii) E.ˆ.u/; ˆ.v// D EQ c .ˆ.u/; ˆ.v// for u 2 F ; v 2 F \ C0 .D/. Here EQ c is the local part of EQ in the Beurling–Deny’s decomposition. Proof. (i) Take f; g 2 C0 .D/ such that suppŒf  \ suppŒg D ;, f D 1 on suppŒu, and g D 1 on suppŒv. Since ˆ.u/ D ˆ.f u/ D ˆ.f /ˆ.u/, ˆ.f / D 1 on ¹x 2 DQ W ˆ.u/ ¤ 0º. On account of (3.3.29), ˆ.f / is a continuous function on DQ by Lemma A.4.3. Hence supp[ˆ.u/] is included in the open set ¹ˆ.f / > 0º. By the same reason, supp[ˆ.v/] is included the open set ¹ˆ.g/ > 0º. Moreover, ¹ˆ.f / > 0º \ ¹ˆ.g/ > 0º D ; because ˆ.f /ˆ.g/ D ˆ.fg/ D ˆ.0/ D 0. Hence suppŒˆ.u/ \ suppŒˆ.v/ D ;. (ii) Suppose that w D k on an open set U . suppŒu/. Take f 2 C0 .D/ such that f D 1 on suppŒu and suppŒf   U . Then ˆ.w/ D k on ¹ˆ.f / > 0º because ˆ.w/ˆ.f / D ˆ.wf / D ˆ.kf / D kˆ.f /. Q FQ / (iii) According to the Beurling–Deny formula, the regular Dirichlet form .E; can be decomposed as Z Q E.u; v/ D EQ c .u; v/ C .e u.x/  e u.y//.e v .x/  e v .y//JQ .dxdy/ Z C

Q D

Q Dnd Q D

Q e u.x/e v .x/k.dx/ for u; v 2 FQ :

We define Radon measures J on D  D n d and k on D as followings: for f; g 2 C0 .D/ with suppŒf  \ suppŒg D ; Z Z f .x/g.y/J.dxdy/ D ˆ.f /.x/ˆ.g/.y/JQ .dxdy/ (3.3.30) DD

Q D Q D

3.3 Maximum Markovian extensions

and for f 2 C0 .D/

Z

Z D

f .x/k.dx/ D

Q D

Q ˆ.f /.x/k.dx/:

139

(3.3.31)

Let E 0 c .u; v/ be the form on F \ C0 .D/ defined by E 0 .u; v/ D EQ c .ˆ.u/; ˆ.v//: c

(3.3.32)

Then E 0 c becomes a strongly local form by Lemma 3.3.6 (ii), and the Dirichlet form E can be decomposed as for u; v 2 F \ C0 .D/ Q E.u; v/ D E.ˆ.u/; ˆ.v//

(3.3.33)

D EQ c .ˆ.u/; ˆ.v// Z .ˆ.u/.x/  ˆ.u/.y//.ˆ.v/.x/  ˆ.v/.y//JQ .dxdy/ C Q Dnd Q D

Z

Q e u.x/e v .x/k.dx/ Z 0c .u.x/  u.y//.v.x/  v.y//J.dxdy/ D E .u; v/ C C

Q D

DDnd

Z C

u.x/v.x/k.dx/: D

Since E D E 0 on C01 .D/, we see from the strongly local property of .E 0 ; DŒE 0 / that J D 0 and k D 0. Hence E.u; v/ D E 0 c .u; v/ and thus Q E.ˆ.u/; ˆ.v// D EQ c .ˆ.u/; ˆ.v//

u; v 2 F \ C0 .D/:

(3.3.34)

Q FQ /. Then Let hui ; chui be the energy measures of the regular Dirichlet form .E; c the equality (3.3.34) implies that hˆ.u/i D hˆ.u/i for u 2 F \ C0 .D/, and thus for v 2 FQ ; u 2 F \ C0 .D/.13 hv;ˆ.u/i D c hv;ˆ.u/i

Lemma 3.3.7. AS i .S /  AM .S /: Q FQ / be its regular representation. Q m; Q E; Proof. Let .E; F / be in AS i .S / and .D; 1 Let ' and  be C0 .D/-function such that ' D 1 on a neighborhood of supp[]. It follows from Lemma 3.3.6 (iii) that for u 2 Fb Q E.u'; / D E.ˆ.u/ˆ.'/; ˆ.// D EQ c .ˆ.u/ˆ.'/; ˆ.//: 13 See

§5.3.

(3.3.35)

140

3 The scope of Dirichlet forms

The right-hand side of (3.3.35) has the integral representation: Z 1 d Q chˆ.u/ˆ.'/;ˆ./i : EQ c .ˆ.u/ˆ.'/; ˆ.// D 2 DQ By the derivation property of Q c and Lemma 3.3.6 (ii), (iii), the right-hand side is equal to Z Z Z 1 1 1 ˆ.u/d Q chˆ.'/;ˆ./i C ˆ.'/d Q chˆ.u/;ˆ./i D d Q chˆ.u/;ˆ./i 2 DQ 2 DQ 2 DQ (3.3.36)

A

D E.u; /:

(3.3.37)

On the other hand, since u' is an element of DŒE 0 , E.u'; / D .u'; S/ dx D .u; S/ dx : Therefore we have E.u; / D .u; S/ dx for all u 2 Fb and all  2 C01 .D/; which implies that C01 .D/ is contained in the domain of self-adjoint operator A associated with .E; F / and that A D S for  2 C01 .D/. Therefore we have Theorem 3.3.2. AM .S / D AS i .S /: On account of Theorem A.4.4, if .E 0 ; DŒE 0 / is conservative, then AS i .S / contains a unique element, .E 0 ; DŒE 0 /. Hence we see from Theorem 3.3.2 that if .E 0 ; DŒE 0 / is conservative, then the Friedrichs extension of S is a unique element of AM .S /. For any regular Dirichlet form, we can consider the class of its Silverstein extensions. It is known14 that the maximum element in the class of Silverstein extensions always exists and it is completely characterized. Example 3.3.1. Let d D 1 and D D .r1 ; r2 / where 1  r1 < r2  1. Let 1 d SD

dx 14 Cf.

  d a ; dx

D.S / D C01 ..r1 ; r2 // :

Z.-Q. Chen and M. Fukushima [1; Theorem 6.6.9].

3.3 Maximum Markovian extensions

141

Then we can easily show that the adjoint operator S  of S is given by   d' 1 d a ; S ' D

dx dx DŒS   D ¹' 2 L2 ..r1 ; r2 /I dx/ W ' is differentiable; ' 0 is absolutely continuous and  d'  2 1 d dx a dx 2 L ..r1 ; r2 /I dx/º: Rx Rx 1 dt , m.x/ D c .t /dt .r1 < c < r2 /. Then S ' is written as Set s.x/ D c a.t / d d' . /. By the same argument as in Example 1.2.2, we d m ds N˛ D ¹0º if and only if both r1 and r2 are not regular. sufficient condition that DŒE C  D DŒE 0 .

can prove that DŒE C  \ This is a necessary and

The next example concerns a case where aij may have some singularity at the origin. Example 3.3.2. Let D D Rd n ¹0º; aij .x/ D jxj2 ıij and .x/ D jxj2 for >  d2 . Then the operator S is written as S ' D 4' C 2 jxj2

n X iD1

xi

@' ; @xi

' 2 C01 .Rd n ¹0º/:

In view of Theorem 3.1.3 the form E.S / .u; v/ D

n Z X iD1

Rd

@u @u 2 jxj dx; @xi @xi

u; v 2 C01 .Rd /;

is closable on L2 .Rd I jxj2 dx/. Let us denote by F the closure of C01 .Rd / with respect to E1 and by Cap the capacity associated with the Dirichlet form .E; F /. Then Cap.¹0º/ > 0 if and only if  d2 < < 1  d2 . Indeed we have seen this for d D 1 in Example 2.2.4. The proof for d > 1 is similar. If  d2 < < 1  d2 , then DŒE 0  is a proper subspace of F . In fact, if these two spaces are identical, then, for f 2 C01 .Rd / with f D 1 on Br D ¹jxj  rº, there exists a sequence ¹fn º  C01 .Rd n ¹0º/ such that E1 .f  fn ; f  fn / ! 0 as n ! 1. Since Cap.¹0º/  E1 .f  fn ; f  fn /, we get Cap.¹0º/ D 0, a contradiction. In particular DŒE 0  differs from DŒE C  because F  DŒE C . On the other hand, if  1  d2 , we have Cap.¹0º/ D 0, and so there exists a sequence ¹fn º  C01 .Rd / such that fn D 1 on B 1 and E1 .fn ; fn / ! 0 n

142

3 The scope of Dirichlet forms

.n ! 1/. We assume that fn ! 0 dx-a.e., by taking a subsequence if necessary. Given u 2 DŒE C b , let un D u  ufn 2 DŒE C b . Then we obtain E1C .u  un ; u  un / D E1C .ufn ; ufn / 2 d Z  X @u  fn jxj2 dx C 2 k u k21 E1 .fn ; fn / ! 0; 2 D @xi

n ! 1:

iD1

Since each function un can be approximated by C01 .Rd n ¹0º/-functions with respect to E1C , we can conclude that DŒE C  D DŒE 0  if  1  d2 . Example 3.3.3. Let D  Rd be a bounded domain with a smooth boundary @D. Let be a C 1 .D/-function satisfying C1 ı.x/"  .x/  C2 ı.x/" for some constants C1 < C2 , where ı.x/ D inf¹jx yj W y 2 @Dº. Let us consider an operator   d 1X @ @' S' D

; ' 2 C01 .D/:

@xi @xi iD1

DŒE C 

Then the spaces and DŒE 0  are identical to the weighted Sobolev space 1 H .DI ı; "/ defined by ° ± @u H 1 .DI ı; "/ D u 2 L2 .DI ı " dx/ W 2 L2 .DI ı " dx/; 1  i  d @xi and the closure of C01 .D/ in H 1 .DI ı; "/ respectively. It is known15 that H 1 .DI ı; "/ D H01 .DI ı; "/ if "  1 or " > 1. But trivially these two spaces differ when " D 0. We now give a necessary and sufficient condition for DŒE C  to be equal to DŒE 0 . Theorem 3.3.3. DŒE C  D DŒE 0  if and only if the Dirichlet form .E C ; DŒE C / satisfies the condition that for any domain O of finite .E C ; DŒE C /-capacity there exists a sequence ¹'n º  C01 .D/ such that (i) 'n ! 1 in measure dx on O, R P n n (ii) limn!1 di;j D1 O aij .x/ @' .x/ @' .x/dx D 0. @x @x i

15 Cf.

A. Kufner [1; §9].

j

3.3 Maximum Markovian extensions

143

Proof. Due to Lemma 3.3.3, it is enough for the proof of sufficiency to show that any function u 2 DŒE C  \ C 1 .D/ can be approximated in DŒE C  by C01 .D/-functions. Without loss of generality, we can assume that u is a bounded non-negative function. Since .u  /C .D u  u ^ / converges to u in E1C by Theorem 1.4.2 (iv) and the capacity of the set ¹u  º is finite by Theorem 2.1.3, we can further assume that u vanishes on the set D n G for some open set G  D of finite .E C ; DŒE C /-capacity. Let us denote a component of the set G by G` and define a sequence ¹u` º1 `D1 by ´ u.x/ for x 2 G` ` u .x/ D 0 for x 2 D n G` : Let ¹'n` º1 nD1 be the sequence of functions specified for the domain G` in the theorem. Then, 'n` u` 2 DŒE 0  and 'n` u` ! u` .n ! 1/ in L2 .DI dx/. Moreover, d Z X

aij

i;j D1 D

2

@ @ .u`  'n` u` / .u`  'n` u` /dx @xi @xj

d Z X i;j D1 G`

.1 

'n` /2 aij

d Z X @u` @u` @' ` @' ` dx C 2 .u` /2 aij n n dx @xi @xj @xi @xj G` i;j D1

! 0 .n ! 1/: P C ` Noting that m `D1 u ! u .m ! 1/ in E1 , we complete the proof of sufficiency. Conversely suppose that DŒE C  D DŒE 0  and let O be an arbitrary domain in D with finite .E C ; DŒE C /-capacity. Take a function u in DŒE C  with u D 1 on O, which can then be approximated in DŒE C  by a sequence ¹'n º1 nD1  1 2 C0 .D/. Since u D 1 on O, 'n ! 1 in L .O; dx/ and d Z X i;j D1 O

aij

d Z X @'n @'n @.'n  u/ @.'n  u/ dx D aij dx ! 0; @xi @xj @xi @xj O

n ! 1;

i;j D1

completing the proof of the necessity. Let us introduce a metric d.S / on D associated to S defined by d.S / .x; y/ D sup¹u.x/  u.y/ W u 2 Dloc ŒE 0  \ C.D/; .u; u/  1 a.e.º; (3.3.38)

144

3 The scope of Dirichlet forms

where .u; u/ D

Pd

@u.x/ @u.x/ 1 i;j D1 .x/ aij .x/ @xi @xj .

We then have16

Theorem 3.3.4. If .D; d.S / / is a complete metric space, then DŒE 0  D DŒE C . In fact, S is essentially self-adjoint, that is, S has a unique self-adjoint extension. Example 3.3.4. Let D D Rd and D 1. Suppose that d X

aij .x/i j  k.jxj C 2/2 .log.jxj C 2//2 jj2

for  2 Rd :

(3.3.39)

i;j D1

Set 1 .x/ D p k We then easily see that

Z

jxj 0

ds : .s C 2/ log.s C 2/

2 Dloc ŒE 0  \ C.Rd / and d X

aij .x/

i;j D1

@ @  1: @xi @xj

Hence, for 8r > 0 ¹x 2 Rd W .0; x/  rº  ¹x 2 Rd W .x/  rº μ ´ Z jxj p ds d  kr : D x2R W .s C 2/ log.s C 2/ 0 Hence .Rd ; d.S / / is complete and thus the uniqueness of Markov or Silverstein extensions holds. Note that if aij .x/ D .jxjC2/2 .log.jxjC2//2 ıi;j , the corresponding diffusion is explosive. This says that the Markov or Silverstein uniqueness can hold even if the minimal diffusion is explosive. It is shown17 that if aij 2 L1 .D/ and D 1, then Markov or Silverstein uniqueness is equivalent to conservativeness of the minimal diffusion. As stated in Example 3.3.1, for one-dimensional diffusion processes, if there exists no exit boundary, then Markov uniqueness is equivalent to conservativeness. Suppose that D D Rd ; D 1 and that aij ; 1  i; j  d , satisfy the uniform ellipticity. Then the domain of finite .E C ; DŒE C /-capacity is necessarily bounded 16 Cf. 17 Cf.

T. Kawabata and M. Takeda [1]. D. W. Robinson and A. Sikora [1].

3.3 Maximum Markovian extensions

145

if d D 1 or 2.18 Therefore, in this case the condition in Theorem 3.3.2 is automatically satisfied. But for d  3 there exists an example that DŒE C  ¤ DŒE 0 .19 On the other hand, Example 3.3.1 implies that DŒE C  D DŒE 0  without the uniform ellipticity if d D 1. But if d D 2, DŒE C  is not identical with DŒE 0  without the uniform ellipticity in general. In fact, we have the following example. Example 3.3.5. Let R2C D ¹.x1 ; x2 / W 0 < x1 < 1; x2 2 R1 º and S ' D 12 4', ' 2 C01 .R2C /. For a fixed C 1 ..0; 1//-function F onto R1 with F 0 .x/ > 0, we define the bijective map  from R2C to R2 by .x1 ; x2 / D .u1 ; u2 / where u1 D F .x1 /;

u2 D

x2 : F 0 .x1 /

Let O be the operator from L2 .R2C I dx/ to L2 .R2 I du/ defined by O W f .x1 ; x2 / 2 L2 .R2C I dx/ ! f . 1 .u1 ; u2 // 2 L2 .R2 I du/: Then the operator O becomes a unitary operator because the Jacobian of  is equal to one. As the inverse operator O 1 of O transforms C01 .R2 / to C01 .R2C /, the symmetric operator S ' D O ı S ı O 1 '; ' 2 C01 .R2 /, is well-defined and written as   2 X @ @' S' D aij @ui @uj i;j D1

where a11 .u1 ; u2 / D .F 0 .F 1 .u1 ///2 ; a12 .u1 ; u2 / D a21 .u1 ; u2 / D u2 F 00 .F 1 .u1 //; a22 .u1 ; u2 / D .u2 F 00 .F 1 .u1 //=F 0 .F 1 .u1 ///2 C .F 0 .F 1 .u1 ///2 : Since det.aij / D 1 and limu1 !1 a11 .u1 ; u2 / D 1, the operator S is not uniformly elliptic. Noting that the operator O satisfies 0  f  1 dx-a.e.

,

O  1 du-a.e.; 0  f

the map A 2 AM .S / ! O ı A ı O 1 is a one to one correspondence between AM .S / and AM .S /, which implies that DŒE C  is not identical with DŒE 0  because the element in AM .S / is not unique. 18 Cf. 19 Cf.

V. G. Maz0 ja [1; §2.7]. V. G. Maz0 ja [1; §2.7].

146

3 The scope of Dirichlet forms

If S of (3.3.17) satisfies the condition (3.3.11), then the closed symmetric form EK corresponding to the Krein extension AK of S possesses the property that N0  DŒEK , EK .u; u/ D 0; 8u 2 N0 , in view of (3.3.12) and (3.3.13). Here N0 D ¹u 2 L2 .DI dx/ W S  u D 0º. Suppose that D is bounded and S D 12 4. Since there is a harmonic function with nonzero finite Dirichlet integral, we see .EK ; DŒEK / ¤ . 12 D; H 1 .D//. Noting that the space .E C ; DŒE C / is nothing but . 12 D; H 1 .D//, we can conclude that AK does not generate a Markovian semigroup because otherwise AK AR and AK D AR . Theorem 3.3.5. If the domain D  Rn is bounded, the semigroup on L2 .D/ generated by the Krein extension AK of the operator 12 4 is not Markovian. We can look upon the family A.S / from a different view point. A 2 A.S / is, in general, a restriction of S  by Lemma 3.3.2 (i). Hence A may be expressed in the following manner: ´ D.A/ D ¹u 2 D.S  / W u satisfies Lº (3.3.40) Au D S  u; u 2 D.A/: Here L stands for the lateral condition to specify the restriction. In many cases, L can be described in terms of the behavior of u near the boundary and L is then called the boundary condition. Let us examine the simplest case that D D .a; b/; 1 < a < b < 1; S u D 1=2u00 ; D.S / D C01 ..a; b//. As we easily see, 8 0º be the semigroup generated by A. Then the following criteria hold22 : T t is positive

,

12  0;

T t is Markovian

,

12  0;

i1 C i 2  0; i D 1; 2:

LR corresponds to ij D 0, while LK corresponds to  11 D  22 D 12 D 1 > 0: Hence the semigroup generated by AK is not even positive. ba

21 W. 22 W.

Feller [1]. Feller [1].

Part II

Symmetric Markov processes

Chapter 4

Analysis by symmetric Hunt processes

A Hunt process is a special Markov process that possesses useful properties such as the right continuity of sample paths, the quasi-left-continuity and the strong Markov property as are introduced in Appendix A.2. In this chapter, we study a symmetric Hunt process in relation to its associated Dirichlet form. From the middle of §4.2 to the end of this chapter except for the last part of §4.4 and §4.7, we shall assume that the associated Dirichlet form is regular. This setting imposes no restriction upon a regular Dirichlet form however, for, as we shall see in §7.2, any regular Dirichlet form admits a Hunt process. In §4.2, some potential theoretic notions for the Dirichlet form appearing in Chapter 2 are interpreted in terms of the Hunt process. In §4.3, 4.4 and 4.6, basic operations in the Hunt process like taking the integrations of functions by hitting distributions, taking the parts of the process on subsets and making ergodic decompositions are linked to certain operations in the Dirichlet space and the extended Dirichlet space. Thus the interplay of the fine topology and quasi topology is a principal character of this chapter. The local property of the Dirichlet form is a necessary and sufficient condition for the Hunt process to possess continuous sample paths with probability 1 (§4.5). §4.5 also justifies our saying that the measures k and J appearing in the Beurling– Deny formula in §3.2 are the killing and jumping measure respectively. Many statements asserted in this chapter end up with the phrase “for q.e. x 2 X”. Under the extra condition of the absolute continuity of the transition function however, one can often strengthen this into “for every x 2 X”, as will be seen at the ends of §4.5 and §4.6. §4.7 formulates an ergodic theorem for a symmetric right process without assuming the regularity of the associated Dirichlet form and its content is independent of the rest of this chapter. In §4.8, we study the Hilbertian structure of the extended Dirichlet space of an irreducible recurrent Dirichlet form probabilistically. Throughout this chapter, we make the convention that any extended real valued function u defined on a subset S  X is extended to S D S [  by setting u./ D 0.

152

4.1

4 Analysis by symmetric Hunt processes

Smallness of sets and symmetry

Let X be a locally compact separable metric space and m be an everywhere dense positive Radon measure on X. We call a Markov process M on .X; B.X// m-symmetric if the transition function of M is m-symmetric in the sense of §1.4. From now on throughout Chapter 4, we fix an m-symmetric Hunt process M D .; M; X t ; Px / on .X; B.X//. We shall use those basic concepts of the Hunt process introduced in Appendix A.2 without repeating explanations. As indicated by the diagram at the end of this section, we study in this section the relationship among several notions of the smallness of sets related to the process M mainly based on the symmetry of M (or rather on the duality with respect to the measure m). The transition function of M is denoted by ¹p t ; t > 0º. The resolvent ¹R˛ ; ˛ > 0º of M is given by  Z 1 Z 1 e ˛t p t .x; E/dt D Ex e ˛t 1E .X t /dt : .4:1:1/ R˛ .x; E/ D 0

0

The kernel R˛ is m-symmetric as well. Both p t and R˛ are uniquely extended to a kernel on .X; B  /; B  being the family of all universally measurable sets of X. B n will denote the family of all nearly Borel sets of X. B n is a subfamily of B  . Lemma 4.1.1. The following conditions are equivalent to each other for E 2 B  : (i) m.E/ D 0, (ii) p t .x; E/ D 0 m-a.e. x 2 X for each t > 0, (iii) R˛ .x; E/ D 0 m-a.e. x 2 X for each ˛ > 0. Proof. RWe only show the implication R (ii))(i). By the m-symmetry of p t ; 0 D lim t #0 X p t .x; E/m.dx/ D lim t #0 E p t 1.x/m.dx/  m.E/: In particular, the above lemma means that any set in B  of potential zero is m-negligible. We know three important notions of exceptional sets in the Hunt probabilistic potential theory – polar set, semi-polar set and set of potential zero. In this book, however, we shall use the term “exceptional set” exclusively in the following sense: e N such a set N  X is called exceptional if thereR exists a nearly Borel set N that Pm . N e < 1/ D 0. Here Pm .ƒ/ D X Px .ƒ/m.dx/; ƒ 2 F1 . Notice that e1 N e still possessing the same properties. we can then find a Borel set N Any exceptional set is m-negligible. In fact, for a Borel exceptional set N we have p t .x; N / D 0 m-a.e. x 2 X and consequently m.N / D 0 by Lemma 4.1.1. A countable union of exceptional sets is also exceptional.

4.1

153

Smallness of sets and symmetry

We say that a set N  X is properly exceptional if N 2 B n , m.N / D 0 and X nN is M-invariant. It is obvious that any properly exceptional set is exceptional. Conversely, it can be shown that any exceptional set is contained in a certain properly exceptional set. Before proving this, we explore some more implications of the m-symmetry of M. Lemma 4.1.2. Em .f0 .X0 /f1 .X t1 /    fn1 .X tn1 /fn .X tn // D Em .fn .X0 /fn1 .X tn tn1 /    f1 .X tn t1 /f0 .X tn // where 0 < t1 <    < tn1 < tn ; f0 ; f1 ; : : : ; fn 2 B C .X/, and Em denotes the integration on  with respect to the measure Pm . Proof. Suppose the above equality holds for a given n. Then Em .f0 .X0 /    fn .X tn /fnC1 .X tnC1 // D Em .f0 .X0 /    .fn  p tnC1 tn fnC1 /.X tn // Z p tnC1 tn fnC1 .x/Ex .fn .X0 /fn1 .X tn tn1 /    f0 .X tn //m.dx/: D X

The last expression equals, by virtue of the m-symmetry of p tnC1 tn , Z fnC1 .x/Ex .EX tnC1 tn .fn .X0 /fn1 .X tn tn1 /    f0 .X tn //m.dx/ X

D Em .fnC1 .X0 /fn .X tnC1 tn /    f0 .X tnC1 //; proving the desired equality of n C 1. For A 2 B n , we put p 0t .x; E/ D Px .X t 2 E; t < A /;

t > 0; x 2 X; E 2 B.X/:

.4:1:2/

This is a Markovian transition function on .X; B  .X// in the sense of §1.4 and indeed the restriction of ¹p 0t ; t > 0º to .X n A; B  .X n A// is the transition function of the part MX nA of the process M on X n A. By making use of Lemma 4.1.2, we can derive the m-symmetry of ¹p 0t ; t > 0º: Lemma 4.1.3. ¹p 0t ; t > 0º is m-symmetric, i.e., Z Z 0 f .x/p t g.x/m.dx/ D g.x/p 0t f .x/m.dx/; X

X

f; g 2 B C .X/:

.4:1:3/

154

4 Analysis by symmetric Hunt processes

Proof. Assume first that A is open. By the previous lemma, we have Z f .x/Ex .g.X t /I Xk t =2n 2 X n A; 1  k  2n ; n D 1; 2; : : :/m.dx/ X Z g.x/Ex .f .X t /I Xk t =2n 2 X n A; 1  k  2n ; n D 1; 2; : : :/m.dx/ D X

for any f; g 2 C0C .X/. Hence, Z Z f .x/Ex .g.X t /I t  A /m.dx/ D g.x/Ex .f .X t /I t  A /m.dx/ X

X

.4:1:4/ in view of the regularity .M:6/ of the sample paths and the closedness of X n A. Replacing t by t C 1=n and letting n ! 1, we arrive at (4.1.3). Let A be R an arbitrary nearly Borel set. Consider a strictly positive function h such that X h.x/m.dx/ D 1. According to Theorem A.2.6, the set A n Ar is semipolar and of potential zero, and consequently m-negligible. In particular, h  m does not charge the set A n Ar . Hence by Theorem A.2.4 there exists a decreasing sequence ¹Gn º of open sets containing A such that Phm .Gn " A ; n " 1/ D 1. Therefore, Ex .g.X t /I t < Gn / increases to Ex .g.X t /I t < A / m-a.e. x 2 X where g  0. Now the identity (4.1.4) for open sets leads us to the same one for the Borel A. Consider the ˛-order hitting distribution HA˛ : HA˛ .x; E/ D Ex .e ˛ A 1E .X A //;

˛ > 0; x 2 X; E 2 B.X/:

.4:1:5/

For a nearly Borel set A; HA˛ is a Markovian kernel on .X; B  .X//. By making use of the strong Markov property, we easily get the so-called Dynkin formula: R˛ f .x/ D R˛0 f .x/ C HA˛ R˛ f .x/;

x 2 X; f 2 B C .X/:

Here ¹R˛0 ; ˛ > 0º is the resolvent of ¹p 0t ; t > 0º defined by   Z A Z 1 0 ˛t 0 ˛t e p t .x; E/dt D Ex e 1E .X t /dt : R˛ .x; E/ D

.4:1:6/

.4:1:7/

0

0

As an immediate consequence of Lemma 4.1.3, we have Corollary 4.1.1. For A 2 B n and ˛ > 0, the identity Z Z ˛ f .x/HA R˛ g.x/m.dx/ D g.x/HA˛ R˛ f .x/m.dx/ X

X

holds for any non-negative universally measurable f and g.

.4:1:8/

4.1

Smallness of sets and symmetry

155

Lemma 4.1.4. Let A be nearly Borel and finely open. If A is m-negligible, then A is exceptional. Proof. Let K be a compact subset of A. By (4.1.8) we have for any f 2 C0 .X/ Z Z ˛ f .x/HK R˛ 1A .x/m.dx/ D HK˛ R˛ f .x/m.dx/ D 0: X

A

Therefore, HK˛ Rˇ 1A .x/ D 0; ˇ > ˛, for m-a.e. x 2 X. Since the hitting distribution HK˛ .x;  / is supported by K and limˇ !1 ˇRˇ 1A .y/ D 1; 8y 2 K  A, we get HK˛ 1.x/ D Ex .e ˛ K / D 0 m-a.e., which means that K is exceptional. R Take now a strictly positive function h such that X h.x/m.dx/ D 1: By Theorem A.2.4., there exists an increasing sequence ¹Kn º of compact subsets of A such that Ehm .e ˛ A / D limn!1 Ehm .e ˛ Kn / D 0: Therefore A is exceptional. We use the term “quasi everywhere” or “q.e.” to mean “except on an exceptional set”. A function u defined q.e. on X is called finely continuous q.e. if there exists a nearly Borel exceptional set N such that X n N is finely open and u is nearly Borel and finely continuous on X n N . Lemma 4.1.5. If u is finely continuous q.e. on X and u  0 m-a.e. on a nearly Borel finely open set E, then u  0 q.e. on E. Proof. Let N be the set appearing in the above definition of q.e. fine continuity. Then the set A D ¹x 2 E n N W u.x/ < 0º is nearly Borel, finely open and m-negligible. Hence, A is exceptional by Lemma 4.1.4. We put for A 2 B n 8

0º on .Rd ; B.Rd // be given by (1.4.20) with a family ¹ t ; t > 0º of probability measures satisfying (1.4.17) and (1.4.19). From the expression Z p t f .x/ D f .x C y/ t .dy/; f 2 C1 .Rd /; .4:1:15/ Rd

it is easy to see that ¹p t ; t > 0º has the Feller property. Let M D .; M; X t ; Px / be the associated Hunt process on .Rd ; B.Rd // according to Theorem A.2.2. We may assume that .!/ D 1; 8! 2 , because Px . < t / D 1  p t 1.x/ D 0, 8x 2 Rd ; 8t > 0: M possesses the spatial homogeneity: Px .X t1 2 E1 ; : : : ; X tk 2 Ek / D P0 .X t1 C x 2 E1 ; : : : ; X tk C x 2 Ek /; t1 <    < tk ; E1 2 B.Rd /; : : : ; Ek 2 B.Rd /, that is, the law of Px is obtained by the spatial shift from the single stochastic process .; M; X t ; P0 / starting at the origin 0. The latter is called a Lévy process, which possesses, together with the regularity (M.6) of sample paths, the property of stationary independent increments. The Hunt process on Rd with the transition function being given by the symmetric stable convolution semigroup  t of index ˛ (see Example 1.5.2) is called the symmetric stable process. In particular, the (standard) Brownian motion on Rd is the Hunt process with  t being given by (1.4.25). This will be seen to be a diffusion at the end of §4.5. M is symmetric with respect to the Lebesgue measure if and only if ¹ t º satisfies (1.4.18). If, moreover,  t is absolutely continuous with respect to the Lebesgue measure (for instance if M is the Brownian motion or the symmetric stable process), then the notion of an exceptional set is merely a synonym for a polar set by Theorem 4.1.2. In this case, any Borel polar set is also properly exceptional. However,  t is not absolutely continuous in general. In the one dimensional case, for example, it is known that  t is purely discontinuous if and only if S D 0 and J is purely discontinuous and bounded in the Lévy–Khinchin formula (1.4.21).2 Thus, the role of the present notion of the exceptional set can not be covered by the traditional notion of the polar set in general (see Theorem 4.2.4). Theorem 4.1.3 is violated by some non-symmetric Markov process. Take the deterministic process M of uniform motion to the right: M has the transition function p t .x; A/ D ı.xCt / .A/; t > 0; x 2 R1 ; A 2 B.R1 / .4:1:16/ Then every singleton is thin but not exceptional. 2 K.

Itô [3; §0.5].

160

4 Analysis by symmetric Hunt processes

4.2

Identification of potential theoretic notions

The transition function ¹p t ; t > 0º of the m-symmetric Hunt process M on X clearly satisfies the continuity condition (1.4.15). Hence, ¹p t ; t > 0º uniquely determines a strongly continuous Markovian semigroup ¹T t ; t > 0º on L2 .XI m/ and a Dirichlet form E on L2 .XI m/ as in Lemma 1.4.3 and the paragraph immediately following it respectively. We call them respectively the semigroup and the Dirichlet form on L2 .XI m/ of the process M. In this section, several probabilistic notions which have appeared in the preceding sections relevant to M are identified with those analytical notions in Chapter 2 relevant to E. First of all, we identify the (1-order) hitting probability pA1 defined by (4.1.9) with the (1-)equilibrium potential eA of Lemma 2.1.1 when A is open. Lemma 4.2.1. If A is an open set with finite capacity, then pA1 is a version of eA . Proof. As T t is the extension to L2 .XI m/ of p t on Bb \ L2 .XI m/, p t f is a version of T t f when f is a universally measurable non-negative function in L2 .XI m/. In particular, if f is ˛-excessive with respect to ¹p t ; t > 0º, then it is also ˛-excessive with respect to ¹T t ; t > 0º in the sense of §2.2. Since pA1 is 1-excessive with respect to ¹p t ; t > 0º and pA1 .x/ D 1; 8x 2 A, it suffices to show, in view of Lemma 2.1.1 and Lemma 2.3.2 that pA1  eA

m-a.e.

.4:2:1/

Take a Borel modification eA of eA such that eA .x/ D 1; 8x 2 A, and let Y t .!/ D e t eA .X t .!//; t  0; ! 2 . Then .Y t ; F tR0 ; Phm / t 0 is a supermartingale for any non-negative Borel function h such that X h.x/m.dx/ D 1. Indeed, the Markov property of M implies, for 0  s < t; Ehm .e t eA .X t /jFs0 / D e s e .t s/ p t s eA .Xs /; Phm -a.s., which in turn is not greater than e s eA .Xs /, Phm -a.s., because the set ¹x 2 XI e .t s/ p t s eA .x/ > eA .x/º is m-negligible. Let D be a finite subset of .0; 1/ with min D D a and max D D b and put .DI A/ D min¹t 2 DI X t 2 Aº. If the set in the braces is empty, we set .DI A/ D b. The optional sampling theorem3 applied to ¹Y t ; F t0 ; Phm º t 2D yields Ehm .e  .DIA/ I .DI A/ < b/  Ehm .Y .DIA/ /  Ehm .Ya /  .h; eA /: Letting D increase to a countable dense subset of .0; b/ and then b tend to infinity, we arrive at .h; pA1 /  .h; eA / which means (4.2.1). 3 Cf.

C. Dellacherie and P. A. Meyer [1; V 11].

4.2 Identification of potential theoretic notions

161

The next theorem justifies the usage of the term “q.e.” in the either sense of §2.1 or §4.2. Theorem 4.2.1. (i) Let ¹An º be a decreasing sequence of open sets with finite capacity. limn!1 Cap.An / D 0 if and only if lim p 1 .x/ n!1 An

D 0 q.e.

.4:2:2/

(ii) Assume that any compact set is of finite capacity. Then a set N  X is exceptional if and only if Cap.N / D 0. Proof. (i) Suppose that Cap.An / ! 0, n ! 1. Since Cap.An / D E1 .eAn ; eAn /  .eAn ; eAn /, we have from Lemma 4.2.1 that limn!1 pA1 n .x/ D 0 m-a.e. which can be strengthen to (4.2.2) by virtue of Lemma 4.1.7. Conversely, by Lemma 4.2.1 and the same reasoning as in the proof of Lemma 2.1.2 (iii), (4.2.2) leads us to limn!1 Cap.An / D limn!1 E1 .pA1 n ; pA1 n / D 0. (ii) Suppose Cap.N / D 0. Then there exist open sets An T N satisfying 1 conditions in (i) and, consequently, pB .x/ D 0 q.e. for B D 1 nD1 An . N /. Hence, N is exceptional. Conversely, assume that N is a compact exceptional set. Let ¹Gn º be a sequence open sets such that Gn G nC1 and T1 of decreasing, relatively compact 1 1 G D N . Then lim p .x/ D pN .x/ D 0 m-a.e. because of (A.2.9) n n!1 nD1 Gn and m-negligibility of N . We have then Cap.N /  limn!1 Cap.Gn / which vanishes as in the proof of the “if” part of (i). Now it suffices to use (2.1.6) to complete the proof. Our next concern is the notions of the continuity of functions introduced in §2.1 and §4.1. Theorem 4.2.2. If u is quasi continuous, then u is finely continuous q.e. More specifically, there exists a properly exceptional set N such that u is nearly Borel measurable on X n N and for any x 2 X n N ,   Px u.X t / is right continuous and lim u.X t 0 / D u.X t  /; 8t 2 Œ0; / D 1; t 0 "t





Px lim u.X t 0 / D u.X  /; X  2 X D Px .X  2 X/: t 0 "

(4.2.3) (4.2.4)

Proof. There exists a decreasing sequence ¹An º1 nD1 of open sets such that limn!1 Cap.An / D 0 and ujX nAn is continuous for each n. Then we have the relation (4.2.2) by Theorem 4.2.1. Denote by N0 the exceptional set in (4.2.2).

162

4 Analysis by symmetric Hunt processes

Due T to Theorem 4.1.1, we can find a properly exceptional set N containing N0 [ . 1 nD1 An /. The property (4.2.3) and (4.2.4) clearly hold for x 2 X n N because   Px lim An D 1 D 1; x 2 X n N: .4:2:5/ n!1

Therefore, u is finely continuous on X n N . In the rest of this section and in the most parts of the subsequent sections of Chapter 4 as well, we need the following assumption (4.2.6) for the process M: the Dirichlet form E on L2 .XI m/ of M is regular:

.4:2:6/

This condition on M is not very restrictive because we show in Chapter 7 that, given any regular Dirichlet form E, there exists uniquely in a certain sense (see Theorem 4.2.7) an m-symmetric Hunt process whose Dirichlet form is E. Notice that the assumption in Theorem 4.2.1 (ii) is fulfilled under (4.2.6). We can easily give a partial converse to Theorem 4.2.2. Lemma 4.2.2. (i) If u is finely continuous q.e. and u 2 F , then u is quasi continuous. (ii) If u is quasi continuous and u 2 F , then u has the property .4:2:3/ with time interval Œ0; / being strengthened to Œ0; 1/. Proof. (i) Let u 2 F be finely continuous q.e. By Theorem 2.1.3, u admits a quasi continuous version e u: e u is then finely continuous q.e. by Theorem 4.2.2. Since u De u m-a.e., we conclude from Lemma 4.1.5 that u D e u q.e. As u differs from a quasi continuous function e u only by a set of zero capacity, u is quasi continuous. (ii) Theorem 2.1.3 specifically means that the function u coincides q.e. with a quasi continuous function in the restricted sense. The restriction that u 2 F in the first assertion of the above lemma will be removed in §4.6. The second assertion will be extended to u 2 Fe in transient case at the end of the next section. The next theorem asserts that the process M is properly associated with its Dirichlet form E in a sense. Theorem 4.2.3. For any non-negative universally measurable function u 2 L2 .XI m/, (i) p t u is a quasi continuous version of T t u; t > 0, (ii) R˛ u is a quasi continuous version of G˛ u; ˛ > 0. We prove this theorem utilizing the following variant of the monotone lemma.

4.2 Identification of potential theoretic notions

163

Lemma 4.2.3. Fix p  1: Let H be a family of non-negative functions on X such that (H.1) u1 ; u2 2 H; c1 u1 C c2 u2  0 for constants c1 ; c2 ) c1 u1 C c2 u2 2 H , (H.2) un 2 H; un " u 2 Lp .XI m/ ) u 2 H , (H.3) 8A open, 9un 2 H; un " 1A . Then H contains all non-negative Borel functions in Lp .XI m/. Proof. Fix a relatively compact open set E  X and put D D ¹A  E W A is Borel and 1A 2 H º. Then D is a Dynkin class containing all open subsets of E and D D B.E/.4 The rest of the proof is clear. Proof of Theorem 4:2:3. We already noticed that p t u is a version of T t u. Since Z 1 Z 1 ˛t e .p t u; v/dt D e ˛t .T t u; v/dt .R˛ u; v/ D 0

0

D .G˛ u; v/;

8v 2 C0 .X/;

R˛ u is a version of G˛ u 2 F . Furthermore, R˛ u is ˛-excessive and, consequently, finely continuous on X. Hence, we get (ii) by virtue of Lemma 4.2.2. We next prove (i) for non-negative u 2 C0 .X/. Then R˛ p t u is a quasi continuous version of G˛ T t u and lim˛!1 ˛R˛ p t u.x/ D lim˛!1 p t .˛R˛ /u.x/ D p t u.x/; 8x 2 X. On the other hand, ˛G˛ T t u is E1 -convergent to T t u 2 F on account of Lemma 1.3.3. Therefore, p t u is a quasi continuous version of T t u by virtue of Theorem 2.1.4 (ii). Let us put H D ¹u 2 L2 .XI m/ W u is non-negative Borel and satisfies (i)º. We have just proved H C0C .X/ and, hence, H satisfies (H.3) of Lemma 4.2.3. (H.1) is trivially satisfied. To check (H.2), let un 2 H increase to u 2 L2 .XI m/. Then T t un is E1 -convergent to T t u by (i) of Lemma 1.3.3. Since p t un is a quasi continuous version of T t un and convergent to p t u pointwise, we get u 2 H by Theorem 2.1.4 (ii) again. Lemma 4.2.3 implies that (i) holds for u Borel. The extension to the universally measurable u is clear. As a result of the regularity of the transition function embodied in Theorem 4.2.3, we can prove the following: Theorem 4.2.4. The equivalent two conditions of Theorem 4:1:2 are also equivalent to the following one: p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X. 4 Cf.

K. Itô [3].

164

4 Analysis by symmetric Hunt processes

Proof. Since this condition implies the second one of Theorem 4.1.2, it is sufficient to derive this property of p t from the first condition (i) of Theorem 4.1.2. Assume that any exceptional set is polar. If A is an m-negligible Borel set, then p t .x; A/ D 0 m-a.e. by Lemma 4.1.1. Since p t .x; A/ is quasi continuous as a function of x 2 X by Theorem 4.2.3, it vanishes q.e. by Lemma 2.1.4. Hence, there exists a Borel polar set N such that p t .x; A/ D 0; 8x 2 X n N . Therefore, p2t .x; A/ D Ex .p t .X t ; A/I X t … N / D 0; 8x 2 X, and we obtain Theorem 4.2.4. We now amplify the first lemma of this section as follows. Let eB be the (1-)equilibrium potential of a set B introduced in Theorem 2.1.5. 1 Theorem 4.2.5. Let B be a nearly Borel set of finite capacity. Then pB is a quasi continuous version of eB .

Proof. For a nearly Borel set B of finite capacity, choose decreasing open sets An B such that limn!1 Cap.An / D Cap.B/. Then eAn is E1 -convergent to a function e0 2 F . Since e0 2 LB and E1 .e0 ; e0 / D Cap.B/, we have that e0 D eB . Hence 1  eB m-a.e. .4:2:7/ pB 1 because pB  pA1 n ; n D 1; 2; : : : : 1 2 F. We first conclude from inequality (4.2.7) and Lemma 2.3.2 that pB 1 1 Lemma 4.2.2 then implies that pB is quasi continuous. Furthermore, pB D 1 1 is a quasi continuous version q.e. on B because of Theorem 4.1.3. Therefore pB of eB by Theorem 2.3.1 (iii).

We state here a 0-order version of Theorem 4.2.3 (ii). We let  Z 1 u.X t /dt : Ru.x/ D Ex

.4:2:8/

0

Theorem 4.2.6. Assume in addition to (4.2.6) that .F R; E/ is transient. For any non-negative universally measurable function u with X uRud m < 1; Ru is a quasi continuous version of Gu .2 Fe /: Proof. In view of the proof of Theorem 1.5.3, there are non-negative functions un 2 L2 .XI m/ such that un " u and Gun 2 Fe are E-convergent to Gu 2 Fe . Moreover, by virtue of the proof of Theorem 1.5.4, Gun is the E-limit of G˛ un 2 F as ˛ # 0. Since Ru is the increasing limit of R˛ un as ˛ # 0 and n ! 1, Theorem 4.2.6 follows from Theorem 4.2.3 (ii) combined with the 0order version of Theorem 2.1.4 and Theorem 2.1.6.

4.2 Identification of potential theoretic notions

165

We say that M satisfies the absolute continuity condition if its transition function fulfills that p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X;

.4:2:9/

which just appeared in Theorem 4.2.4. This condition and its consequences stated below are often useful in strengthening “quasi everywhere” statements into “everywhere” ones. Exercise 4.2.1. Assume that ¹p t º is strongly Feller: p t .Bb .X/  Cb .X/. Show that M then satisfies the absolute continuity condition. Lemma 4.2.4. Assume the absolute continuity condition. There exists then a nonnegative jointly measurable function r˛ .x; y/; ˛ > 0; x; y 2 X, such that Z r˛ .x; y/f .y/m.dy/; x 2 X; f 2 Bb .X/; (4.2.10) R˛ f .x/ D X

(4.2.11) r˛ .x; y/ is ˛-excessive in x (and in y); r˛ .x; y/ D r˛ .y; x/; Z r˛ .x; z/rˇ .z; y/m.dz/; ˇ > ˛ > 0: r˛ .x; y/ D rˇ .x; y/ C .ˇ  ˛/ X

(4.2.12) Proof. We can find a non-negative jointly measurable function, say v˛x .y/, satisfying (4.2.10). Fixing ˛ > 0 and x 2 X, we regard v˛x .y/ as a function of y. By the symmetry of p t , we have e ˛t .p t v˛x ; f / D e ˛t .v˛x ; p t f / D e ˛t R˛ p t f .x/;

.4:2:13/

which is not greater than R˛ f .x/ D .v˛x ; f / for any non-negative f and hence e ˛t p t v˛x .y/  v˛x .y/ for m-a.e. y 2 X: By virtue of the absolute continuity of p t , we see then that e ˛t p t v˛x .y/ is, for every fixed y 2 X, non-decreasing as t # 0. It suffices to let r˛ .x; y/ D lim e ˛t p t v˛x .y/: t #0

Indeed r˛ .x; y/ is then ˛-excessive in y and, by letting t # 0 in (4.2.13), we get (4.2.10) for f 2 CbC .X/. Further r˛ .x; y/ is an increasing limit as t # 0 R y y of e ˛t X p t .y; dz/v˛x .z/ D R˛ g t .x/ where g t .z/ D e ˛t p t .y; dz/=m.dz/. y Since R˛ g t .x/ is ˛-excessive in x, so is r˛ .x; y/. The proof of (4.2.11) and (4.2.12) is now easy.

166

4 Analysis by symmetric Hunt processes

Exercise 4.2.2. Assume the absolute continuity condition and consider the resolvent density appearing in the lemma above. Show that, for a positive Radon measure  on X,  is of finite energy integral, i.e.,  2 S0 , if and only if Z r1 .x; y/.dx/.dy/ < 1: X X

Show also that in this case the function Z R˛ .x/ D r˛ .x; y/.dy/

.4:2:14/

X

is a quasi continuous and ˛-excessive version of the ˛-potential U˛  of ; ˛ > 0. We shall be also concerned with a condition on M slightly weaker than (4.2.9): p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X n N ;

.4:2:15/

where N is a certain Borel properly exceptional set. In other words, this condition says that there exists ˇa Borel properly exceptional set N  X such that the restricted Hunt process MˇX nN satisfies the absolute continuity condition (4.2.9). Theorem 4.2.7. Suppose the Dirichlet form E satisfies the Sobolev type inequality: for some q > 2 and a positive constant S; 2 kukL q .X Im/  S E1 .u; u/;

u2F:

.4:2:16/

Then M satisfies (4.2.15) for some Borel properly exceptional set N: Proof. Let ¹T t I t > 0º be the semigroup on L2 .XI m/ associated with the Dirichlet form E: It is known5 that the Sobolev type inequality (4.2.16) is equivalent to the inequality for the semigroup kT t f kL1 .X Im/  C e t t =2 kf kL1 .X Im/ ;

t > 0; f 2 L1 .XI m/; .4:2:17/

2q : holding for a positive constant C and  D q2 C Choose a countable subfamily H0 of C0 .X/ such that, for any open set A  X; there exists ¹fn º  H0 with fn " 1A ; n ! 1: On account of Theorem 2.1.2 and Theorem 4.2.3, we can find a regular nest ¹Fk º such that

¹p t f W f 2 H0 ; t 2 QC º  C1 .¹Fk º/: 5 Cf.

E. A. Carlen, S. Kusuoka and D. W. Stroock [1].

4.2 Identification of potential theoretic notions

167

F : Suppose (4.2.16), then we see from (4.2.17) that, for We let N D X n [1 kD1 k each x 2 X n N; the inequality Z t =2 f .x/m.dx/ .4:2:18/ p t f .x/  C e t X

holds true for any t 2 QC and any f 2 H0 : (4.2.18) extends to any t > 0; and then to any non-negative Borel f 2 L1 .XI m/ by Lemma 4.2.3, yielding the absolute continuity of p t .x;  / with respect to m for each t > 0 and x 2 X n N: So far we have fixed an m-symmetric Hunt process M on X whose Dirichlet form E on L2 .XI m/ is regular. Theorem 4.2.3 enables us to formulate a statement concerning the uniqueness of such a process. Let us say that two m-symmetric Hunt processes are equivalent if they possess a common properly exceptional set outside which their transition functions coincide. Theorem 4.2.8. Let M1 and M2 be two m-symmetric Hunt processes on X possessing a common regular Dirichlet space on L2 .XI m/. Then M1 and M2 are equivalent. .i /

Proof. Denote by p t the transition function of Mi ; i D 1; 2: By virtue of Theorem 4.2.3 and Lemma 2.1.5, there exists a set B0 of capacity zero such that .1/

.2/

p t u.x/ D p t u.x/;

8x 2 X n B0

.4:2:19/

for every t 2 QC and u 2 C1 .X/; C1 .X/ being a countable uniformly dense subset of C0 .X/. (4.2.19) then holds for every t > 0 by right continuity. Applying Theorem 4.1.1 and Theorem 4.2.1 to M1 and M2 alternatively, we can find an increasing sequence ¹Bn º of Borel sets containing B0 such that SB2n1 (resp. B2n ) is properly exceptional relative to M1 (resp. M2 ). Then B D 1 nD1 Bn is a common .1/ .2/ properly exceptional set outside which p t D p t ; 8t > 0. Example 4.2.1. Let M be the spatially homogeneous symmetric Markov process on Rd (see Example 4.1.1). Then the Dirichlet form of M on L2 .Rd / is given by (1.4.24) and M satisfies the condition (4.2.6) according to Example 1.4.1. The d -dimensional Brownian motion is a special example of M and its Dirichlet form is . 12 D; H 1 .Rd //, H 1 .Rd / being the Sobolev space of order 1. The Dirichlet form E of the transient symmetric stable process M of index ˛ is given by (1.4.24) with .x/ D cjxj˛ ; 0 < ˛  2; ˛ < d; c > 0. On account of Theorem 2.1.6 and Example 2.2.2, a set N  Rd is of zero capacity with respect to E1 if and only if the set N is of zero capacity based on the Riesz

168

4 Analysis by symmetric Hunt processes

kernel R.˛/ . Therefore Theorem 4.2.1 and Example 4.1.1 imply that a set N is of zero capacity based on the Riesz kernel if and only if N is polar with respect to the stable process M. Thus, the Brownian polar set is just the set of Newtonian capacity zero when d  3. When d D 2, the Brownian polar set is just the set of logarithmic capacity zero in view of Example 2.2.3.6 Since the Brownian paths X t are continuous (see the paragraph after Theorem 4.5.4), it follows from Theorem 4.2.2 that any quasi continuous function u on Rd has the following property: Px .u.X t / is continuous in t 2 Œ0; 1// D 1;

8x 2 Rd n N;

N being a certain Borel polar set depending on u. Theorem 4.2.4 is violated by the uniform motion to the right: it has the transition function (4.1.16) which is not absolutely continuous with respect to Lebesgue measure. But the resolvent R˛ .x; dy/ has the density ´ e ˛.xy/ x < y R˛ .x; y/ D 0 x  y:

4.3

Orthogonal projections and hitting distributions

We still consider the m-symmetric Hunt process M on X satisfying condition (4.2.6). Just as in §2.3 we set for a fixed nearly Borel set B  X FX nB D ¹u 2 F W e u D 0 q.e. on Bº;

.4:3:1/

e u being a quasi continuous version of u: FX nB is a closed subspace of the Hilbert space .F ; E˛ / for each ˛ > 0. Denote by HB˛ its orthogonal complement: F D FX nB ˚ HB˛ :

.4:3:2/

The orthogonal projection on the space HB˛ is denoted by PHB˛ . The ˛-order hitting distribution HB˛ has been defined by (4.1.5). In dealing with a function v defined and non-negative q.e. on X, we always assume that v is extended to be a non-negative universally measurable function on X. Then the operation HB˛ v.x/ makes sense for every x 2 X and HB˛ v.x/ D Ex .e ˛ B v.X B //. The key theorem of this section is the following: Theorem 4.3.1. For each u 2 F and ˛ > 0; HB˛ je uj.x/ is finite q.e. and HB˛e u is a quasi continuous version of PHB˛ u. 6 This

goes back to S. Kakutani [1] and J. L. Doob [1].

4.3 Orthogonal projections and hitting distributions

169

First we shall prove a lemma. Lemma 4.3.1. Let u 2 F be an ˛-excessive function with respect to ¹p t ; t > 0º. Denote by uB the ˛-reduced function of u on B defined in 2:3. Then HB˛ u is a quasi continuous version of uB . Proof. As we saw in the preceding section, u is quasi continuous and ˛-excessive with respect to ¹T t ; t > 0º. It suffices to show HB˛ u  uB HB˛ u.x/

m-a.e.;

(4.3.3)

D u.x/ q.e. on B:

(4.3.4)

In fact HB˛ u being ˛-excessive and finely continuous, we see from (4.3.3) and Lemma 2.3.2 that HB˛ u 2 F . Then Lemma 4.2.2 and (4.3.4) lead us to HB˛ u D HB˛ u D u D e u q.e. on B. Thus, we can conclude HB˛ u D uB by Theorem 2.3.1. Since (4.3.4) follows immediately from Theorem A.2.6 and Theorem 4.1.3, it only remains to verify the inequality (4.3.3). To prove (4.3.3) we take a non-negative Borel quasi continuous version e uB of uB . We claim that there exists a properly exceptional set N such that

A

uB .x/  e uB .x/; e ˛t p t e e uB .x/ D u.x/;

8t > 0; 8x 2 X n N;

(4.3.5)

8x 2 B n N;

(4.3.6)

uB .X t / is right continuous in t  0/ D 1; Px .e

x 2 X n N:

(4.3.7)

uB is quasi continuous Since uB is ˛-excessive with respect to ¹T t ; t > 0º and p t e by Theorem 4.2.3, we see that (4.3.5) holds q.e. x 2 X for each t > 0. By (2.3.4), (4.3.6) is true for q.e. x 2 B. Furthermore, Theorem 4.2.2 asserts the validity of (4.3.7) for q.e. x 2 X. According to Theorem 4.1.1, we can choose a properly exceptional set N such that the inequality (4.3.5) for every rational t > 0, the equality (4.3.6) and the relation (4.3.7) hold for all x 2 X n N . An application of Fatou’s lemma then yields (4.3.5) for every t > 0. Fix x 2 X n N . Then (4.3.5) and (4.3.7) imply that the stochastic process ˛t .e e uB .X t /; F t ; Px / is a right continuous non-negative supermartingale. The optional sampling theorem then gives HB˛e uB .x/ D Ex .e ˛ B e uB .X B //  e uB .x/. ˛ uB .x/ D HB˛ u.x/ from (4.3.6) and (4.3.7). We On the other hand, we have HB e get HB˛ u  e uB q.e. proving (4.3.3). Proof of Theorem 4:3:1. Suppose first that u 2 F is ˛-excessive with respect to ¹p t ; t > 0º. Then the reduced function uB of u on B coincides with PHB˛ u as we saw already in §2.3. Hence HB˛ u is a quasi continuous version of PHB˛ u by virtue of Lemma 4.3.1.

170

4 Analysis by symmetric Hunt processes

In dealing with a general non-negative u 2 F , we may assume that u is bounded because, otherwise, the approximation of u by the truncations u ^ n works. For u can be then expressed as a difference of two bounded ˛-exceseach ˇ > 0; Rˇ e u / is a quasi continuous version of sive functions. Hence, HB˛ .Rˇ e PHB˛ .Gˇ u/. On the other hand, since ˇGˇ u is E1 -convergent to u as ˇ ! 1 on account of Lemma 1.3.3, so is PHB˛ .ˇGˇ u/ to PHB˛ u. Furthermore, by choosing a properly exceptional set N such that the relation (4.2.3) holds for e u and u /.x/ D HB˛e u.x/; x 2 X n N . Due to x 2 X n N , we can get limˇ !1 HB˛ .ˇRˇ e Theorem 2.1.4, we conclude that HB˛e u is a quasi continuous version of PHB˛ u. It is not difficult to derive from Theorem 4.3.1 its 0-order counterpart. We assume the transience of .E; F /. Then its extended Dirichlet space Fe is a Hilbert space and admits, for a nearly Borel set B  X, the orthogonal decomposition Fe D Fe;X nB ˚ HB ; Fe;X nB D ¹u 2 Fe W e u D 0 q.e. on Bº:

(4.3.8)

The orthogonal projection on HB is denoted by PHB . The (0-order) hitting distribution is defined by HB .x; E/ D Px .X B 2 E I B < 1/: Then HB v.x/ D Ex .v.X B / I B < 1/: uj.x/ is Theorem 4.3.2. Assume the transience of .E; F /. For each u 2 Fe ; HB je u is a quasi continuous modification of PHB u. finite q.e. and HBe We prepare a lemma. Lemma 4.3.2. Assume the transience of .E; F /. If u 2 Fe is bounded, then HBe u is a quasi continuous element of Fe and there exist un 2 F and ˛n # 0 such that kHB˛ne un  HBe ukE ! 0;

kun  ukE ! 0;

n ! 1;

.4:3:9/

and un ; HB˛ne un / < 1: sup ˛n .HB˛ne

.4:3:10/

n

Proof. Suppose u 2 Fe satisfies juj  M for some constant M . Take un 2 F which is E-convergent to u. On account of the 0-order version of Theorem 1.4.2 and Theorem 2.1.4, we may assume that jun j  M and e un ! e u q.e.

4.3 Orthogonal projections and hitting distributions

171

Put Cn D .un ; un / and choose ˛n # 0 such that ˛n Cn  1. By virtue of Theorem 4.3.1, un ; HB˛ne un /  E˛n .un ; un /  E.un ; un / C 1: E˛n .HB˛ne Therefore both E.HB˛ne un ; HB˛ne un / and ˛n .HB˛ne un ; HB˛ne un / are uniformly un is E-conbounded. In particular, the Cesàro mean of a subsequence of HB˛ne ˛n vergent. But HB e un converges to HBe u q.e. by the bounded convergence theorem. Hence by the 0-order version of Theorem 2.1.4, we get the lemma by replacing un by the above Cesàro mean. Proof of Theorem 4:3:2. Suppose first that u 2 Fe is bounded. It suffices to show that u De u q.e. on B; HBe u; v/ D 0; E.HBe

8v 2 Fe;X nB :

(4.3.11) (4.3.12)

(4.3.11) is immediate from Theorem 4.1.3. To see (4.3.12), take un 2 F of Lemma 4.3.2 and observe that un ; v/ D 0; E˛n .HB˛ne

8v 2 FX nB :

Because of (4.3.10), q ˇ ˇ q q ˛n ˛n ˇ˛n .H ˛ne ˇ ˛n ˛n .HB e un ; HB e un / .v; v/ ! 0; B un ; v/ 

n ! 1;

and we get E.HBe u; v/ D 0;

8v 2 FX nB :

.4:3:13/

By Lemma 4.3.2 again, any v 2 Fe;X nB admits a sequence vn 2 F and ˛n # 0 such that vn  HB˛n vn 2 FX nB is E-convergent to v  HB v D v. Hence (4.3.13) extends to (4.3.12). For a general non-negative u 2 Fe ; un D u ^ n 2 Fe is E-convergent, HBe un 2 HB is E-convergent and HBe un " HBe u pointwise. Hence HBe u is finite q.e. and a quasi continuous modification of PHB u. We can now prove the 0-order counterpart of Theorem 4.2.5. The (0-order) hitting probability of a set B is given by pB .x/ D Px .B < 1/:

.4:3:14/

Theorem 4.3.3. Assume the transience of .E; F /. Let B be a nearly Borel set of .0/ finite 0-order capacity and eB be its 0-order equilibrium potential. Then pB is .0/

a quasi continuous modification of eB .

172

4 Analysis by symmetric Hunt processes .0/

.0/

Proof. By virtue of the 0-order version of Theorem 2.1.5, eB D PHB eB . Hence .0/ .0/ eB is a quasi continuous modification of eB by the preceding theorem. Since HBe .0/

.0/

eB is right continuous along the sample path (Theoe eB D 1 q.e. on B and e rem 4.2.2), .0/

eB .X B / D 1; B < 1/ D Px .B < 1/ q.e. Px .e .0/

Consequently HBe eB D pB q.e. Corollary 4.3.1. Assume the transience of .E; F /. (i) For a decreasing sequence ¹An º of open sets, lim Cap.0/ .An / D 0 if and only if

n!1

lim pAn .x/ D 0 q.e.

n!1

(ii) If u is quasi continuous and u 2 Fe , then u satisfies (4.2.3) with Œ0; / being replaced by Œ0; 1/. Proof. (i) follows from Theorem 4.3.3 and the transience inequality (1.5.8). (ii) In view of the 0-order version of Theorem 2.1.3, u is equal q.e. to a quasi continuous function in the restricted sense relative to Cap.0/ . We can then use (i) in a similar way as in the proof of Theorem 4.2.2 to get the strengthened version of (4.2.3). Theorem 4.3.2 will be extended to the general (not necessarily transient) case in §4.6.

4.4

Parts of forms and processes

We continue with the setting of the preceding section; M is an m-symmetric Hunt process on X whose Dirichlet form E on L2 .XI m/ is regular. X nB For ˛ > 0 and a nearly Borel set B  X, let us introduce the kernel R˛ by   Z B X nB ˛s e 1E .Xs /ds ; x 2 X; E 2 B.X/: .4:4:1/ R˛ .x; E/ D Ex 0

X nB

As we already saw in §4.1, ¹R˛ ; ˛ > 0º is an m-symmetric resolvent kernel on .X; B  .X// and the following Dynkin formula holds: R˛ f .x/ D R˛X nB f .x/ C HB˛ R˛ f .x/;

x 2 X; f 2 B C .X/:

.4:4:2/

4.4

Parts of forms and processes

173

For ˛ > 0 and f 2 B C .X/\L2 .XI m/, this formula expresses the orthogonal decomposition (4.3.2) of R˛ f 2 F by virtue of Theorem 4.3.1 and Theorem 4.2.3. X nB R0C ; R0C and HB0C are denoted by R; RX nB and HB respectively. Thus Rf .x/ D RX nB f .x/ C HB Rf .x/;

x 2 X; f 2 B C .X/: .4:4:3/ R If .E; F / is transient, then, for f 2 B C .X/ such that X f Rf d m < 1, formula (4.4.3) represents the orthogonal decomposition (4.3.8) of Rf 2 Fe on account of Theorem 4.3.2 and Theorem 4.2.6. In particular we have arrived at the following theorem. X nB

Theorem 4.4.1. (i) If ˛ > 0 and f 2 B C .X/\L2 .XI m/, then R˛ X nB continuous, R˛ f 2 FX nB and E˛ .R˛X nB f; v/ D .f; v/;

8v 2 FX nB :

f is quasi

.4:4:4/

R (ii) Suppose that .E; F / is transient. If f 2 B C .X/ satisfies X f Rf d m < 1, then RX nB f is quasi continuous, RX nB f 2 Fe;X nB and E.RX nB f; v/ D .f; v/;

8v 2 Fe;X nB :

.4:4:5/

The restriction of the Dirichlet form E to FX nB FX nB is denoted by .FX nB ; E/: FX nB is not necessarily dense in L2 .XI m/, but clearly .FX nB ; E/ is a Dirichlet form on L2 .XI m/ in the wide sense (see §1.4). According to Theorem 1.4.3, this is uniquely related to a Markovian resolvent on L2 .XI m/ which is not necessarily strongly continuous. Theorem 4.4.1 means that this resolvent is determined just X nB by ¹R˛ ; ˛ > 0º. Let us denote the set X n B by G. The set FG is a subspace of L2G .XI m/ D ¹u 2 L2 .XI m/ W u D 0 m-a.e. on Bº which can be identified with L2 .GI m/ in an obvious manner. Hence, the form .FG ; E/ can also be considered as a Dirichlet form on L2 .GI m/ in the wide sense. We denote this by EG and call it the part of the Dirichlet form E on the nearly Borel set G. The part MG of the Markov process M on the nearly Borel G has been introduced in §A.2 and, indeed, MG is a Markov process on .G; B.G// with an m-symmetric transition function p G t on .G; B.G// defined by (4.1.2) for A D B. But MG is not normal unless G is finely open. Theorem 4.4.2. Assume that m-almost all points of G are irregular for B D X nG: m.G \ B r / D 0:

.4:4:6/

174

4 Analysis by symmetric Hunt processes

Then EG is a Dirichlet form on L2 .GI m/. EG is associated with MG in the sense that the strongly continuous semigroup ¹T tG ; t > 0º on L2 .GI m/ corresponding to EG is determined by the transition function ¹p G t ; t > 0º of MG . r G Proof. Since lim t #0 p G t u.x/ D u.x/; 8x 2 G n B for any u 2 C0 .X/, ¹p t ; t > 0º determines a strongly continuous semigroup ¹T tG ; t > 0º on L2 .GI m/ on account of Lemma 1.4.3. Theorem 4.4.1 reads as follows:

R˛G f 2 FG

and

EG;˛ .R˛G f; v/ D .f; v/L2 .GIm/ ;

8v 2 FG ;

for any f 2 B C .X/ \ L2 .GI m/. Since the resolvent of MG is the restriction to .G; B.G// of ¹R˛G ; ˛ > 0º, this implies that EG is the Dirichlet form on L2 .GI m/ associated with the strongly continuous semigroup ¹T tG ; t > 0º. Condition (4.4.6) is satisfied if G is finely open q.e. (§4.6) and in particular G is open. When G is open, the part MG of M on G is a Hunt process on .G; B.G// (see A.2). The part EG of the form E on G also has a nicer property: Theorem 4.4.3. Let G be an open set. (i) The part EG of the Dirichlet form E on G is a regular Dirichlet form on L2 .GI m/. More specifically, for any special standard core C of E, the space CG defined by .2:3:12/ is a core of EG . (ii) A subset of G is of capacity zero with respect to EG if and only if it is of capacity zero with respect to E. A function is quasi continuous on G with respect to EG if and only if it is quasi continuous with respect to E. G C 2 (iii) p G t f is a quasi continuous version of T t f for any f 2 B .G/\L .GI m/.

Proof. (i) Let C be a special standard core of E. Clearly CG is uniformly dense in C0 .G/. CG is also E1 -dense in FG by Lemma 2.3.4. Hence EG is regular. (ii) Denote by CapG .A/ the 1-capacity of an open set A  G with respect to EG . Let ¹An º be a sequence of decreasing open subsets of G. It is enough to show the equivalence CapG .An / # 0 , Cap.An / # 0. The implication “)” is clear from the obvious inequality CapG .A/  Cap.A/ for open A  G. Since EG is the Dirichlet form of the process MG , the proof of Lemma 4.2.1 applies and we see that the 1-equilibrium potential eAG with respect to EG possesses as its version the (1-)hitting probability pAG .x/ D Ex .e  A I A < G / with respect to the process MG . If Cap.An / # 0, then pAn .x/ # 0 m-a.e. and, thus, pAGn .x/ # 0 m-a.e. Hence, CapG .An / D EG;1 .pAGn ; pAGn / ! 0, n ! 1. (iii) follows from (i), Theorem 4.4.2 and Theorem 4.2.3.

4.4

Parts of forms and processes

175

Theorem 4.4.4. Suppose .E; F / is transient and let G be a nearly Borel set satisfying .4:4:6/. Then the part EG of E on G is transient and its extended Dirichlet space equals .Fe;G ; E/. Proof. Let g be a reference function of the transient Dirichlet space .E; F /. Obviously EG is then transient with a reference function gG the restriction of g to G. By virtue of Theorem 4.4.1 (ii), we have for any f RD h  gG with non-negative 1 bounded measurable function h on G that RG f .D 0 p G t f dt / 2 Fe;G and E.RG f; v/ D .f; v/;

8v 2 Fe;G :

Therefore Theorem 1.5.5 implies that .Fe;G ; E/ is the extended Dirichlet space of EG . Exercise 4.4.1. Let E be a countable set with discrete topology and M D .X t ; Px / be an m-symmetric Hunt process on E associated with the regular Dirichlet form .E; F 0 / of Example 1.2.5. Denote by i the first exit time of M from i 2 E: i D inf¹t > 0 W X t … iº: Show that i is exponentially distributed under Pi with Ei Œi  D 

mi ; qi i m0i

i 2 E;

and, for each i 2 E; Pi .Xi D j / D 

qij ; qi i

j ¤ i;

Pi .Xi D / D 

X qij ki D1C : qi i qi i j ¤i

For the sake of later use, we finally consider a general right process M on X which is m-symmetric. The associated Dirichlet form on L2 .XI m/ is denoted by .E; F / whose regularity is not assumed however. The part MG of M on a nearly Borel set G has the resolvent ¹R˛G ; ˛ > 0º defined by (4.4.1) for B D X n G: We show that the form E still serves to describe MG : We use the term ‘q.e.’ to mean ‘except for an exceptional set’ for M in the sense of §4.1. Theorem 4.4.5. (i) Assume that G is either an open set or the fine interior of a nearly Borel finely closed set. The transition function of MG is then m-symmetric and generates a strongly continuous Markovian semigroup on L2 .GI m/: The bG / of MG on L2 .GI m/ satisfies the following: F bG  Dirichlet form .b EG; f b b b F and E G D E on F G  F G : S (ii) Let ¹Gn º be an increasing sequence of subsets of X as in (i) such that 1 nD1 Gn S1 b D X q.e. Then nD1 F Gn is E1 -dense in F .

176

4 Analysis by symmetric Hunt processes

Proof. It is enough to show for a set G as in (i) that R˛G .C0 .X//  F ;

E˛ .R˛G f; R˛G g/ D .R˛G f; g/;

f; g 2 C0 .X/:

.4:4:7/

Indeed this implies the m-symmetry of the transition function of MG ; which in turn generates a strongly continuous Markovian semigroup on L2 .GI m/ as Theorem 4.4.2. The right-hand side of (4.4.7) equals b E G;˛ .R˛G f;R˛G g/ and R˛G .C0 .X// b G yielding the conclusion in (i). Furthermore, for a sequence ¹Gn º is dense in F as in (ii) and for any f 2 C0 .X/, E˛ .R˛ f  R˛Gn f; R˛ f  R˛Gn f / D .f; R˛ f /  .f; R˛Gn f / ! 0;

n ! 1;

because R˛Gn f ! R˛ f q.e. as n ! 1; due to the quasi-left-continuity of X: This means that (ii) is true. We first prove (4.4.7) with G replaced by a nearly Borel finely closed set F . We let E D X n F and, for a fixed f 2 C0 .X/,  Z 1 Rt E ˛t  0 1E .Xs /ds R˛ f .x/ D Ex e f .X t /dt : 0

E

We have then R˛F f .x/ D lim !1 R˛ f .x/ because E is finely open and E D Rt inf¹t > 0 W 0 1E .Xs /ds > 0º: Consequently E.R˛F f; R˛F f / D lim

lim ˇ.R˛E f  ˇRˇ R˛E f; R˛F f /:

ˇ !1  !1

.4:4:8/

This identity holds in the sense that, if the limit in the right-hand side is finite, then R˛F f 2 F and the limit coincides with the left-hand side. To compute the right-hand side, we utilize the generalized resolvent equation Rˇ f  R˛E f C .ˇ  ˛/Rˇ R˛E f  Rˇ .1E  R˛E f / D 0:

.4:4:9/

This equation for ˇ D ˛ implies that R˛E f 2 F ; E˛ .R˛E f; v/ C .R˛E f; v/1E m D .f; v/:

.4:4:10/

E

In particular, R˛ is m-symmetric. Keeping this in mind, we can rewrite the righthand side of (4.4.8) as lim

lim ˇ.Rˇ f  ˛Rˇ R˛E f  Rˇ .1E  R˛E f /; R˛F f /

ˇ !1  !1

D .f; R˛F f /  ˛.R˛F f; R˛F f /  lim

lim .f; R˛E .1E  ˇRˇ R˛F f //:

ˇ !1  !1

4.4

Parts of forms and processes

177

If we introduce the random functional  t by ² ³ Z s 1E .Xv /dv > t ;  t D inf s > 0 W 0

Z

then R˛E .1E

 g/.x/ D Ex

1

e

˛ t  t

 g.X t /dt ;

0

which tends to Ex .e ˛0 g.X0 // as ! 1 provided that g is bounded and finely continuous. Since 0 D E and Px .X0 2 E r / D 1 by Lemma A.2.7 and further R˛F f .x/ D 0 for x 2 E r , we get that lim

lim .f; R˛E .1E  ˇRˇ R˛F f // D Ef m .e ˛0 R˛F f .X0 // D 0:

ˇ !1  !1

Thus we arrive at (4.4.7) for G replaced by a nearly Borel finely closed set F . In the same way, we also have the relation 0

E˛ .R˛F f; R˛F f / D .f; R˛F f / if F and F 0 are nearly Borel finely closed and F  F 0 . Assume now that G is the fine interior of a nearly Borel finely closed set F : G D ¹x 2 X W Px .E > 0/ D 1º; E D F c : Then G is a nearly Borel finely open set and B.D G c / D E r ; which readily implies that Px .B D E / D 1; x 2 X; in view of Lemma A.2.7. Therefore (4.4.7) for G and that for F are the same. Take finally any open set G and a sequence ¹Fn º of closed sets increasing to G. Denoting R˛Fn f by R˛n f , we get, for m < n; E˛ .R˛n f  R˛m f; R˛n f  R˛m f / D .f; R˛n f /  .f; R˛m f /, which means that ¹R˛n f º is E˛ -Cauchy. Since limn!1 R˛n f .x/ D R˛G f .x/ by the quasi-left-continuity of X, we see that (4.4.7) is valid for the open set G: Exercise 4.4.2. Show the equation (4.4.9). Example 4.4.1. Let M be the d -dimensional Brownian motion. According to Example 1.4.1, the Dirichlet form E of M is given by ( 12 D; H 1 .Rd //. Let D be a domain in Rd . According to the identity (2.3.19) in Example 2.3.1, the part on D of the Dirichlet form ( 12 D; H 1 .Rd // is nothing but the form ( 12 D; H01 .D// on L2 .D/. By virtue of Theorem 4.4.3, the form ( 12 D; H01 .D// is the Dirichlet form of the part of M on the domain D, which is often called the absorbing Brownian motion on D. We can also conclude from (2.3.18) and Theorem 4.3.1 that a non-negative u D HR˛ d nDe u. function u 2 H 1 .Rd / is ˛-harmonic on D if and only if e

178

4 Analysis by symmetric Hunt processes

Example 4.4.2. Consider the half-space D  Rd , i.e., D D ¹x D .x 0 ; xd / W x 0 2 Rd 1 ; xd > 0º. We set @D D ¹x D .x 0 ; xd / W x 0 2 Rd 1 ; xd D 0º and D D D [ @D. The Sobolev space ( 12 D; H 1 .D// is a Dirichlet space on L2 .D/ which is not regular (see Example 1.2.3). However if we regard this as a Dirichlet space on L2 .D/ rather than on L2 .D/, then this is regular as was seen in Example 1.5.3 and Example 1.6.2. b D .X b t ; Px / The associated diffusion M x2D on D is the reflecting Brownian b motion; i.e., M is obtained from the Brownian motion M D .X t ; Px /x2Rd on Rd by setting b t D .X t0 ; jX t.d / j/; where X t D .X t0 ; X t.d / /: .4:4:11/ X b is expressed by the b˛ of M To verify this assertion, observe that the resolvent R resolvent R˛ of M as follows: b˛ f .x/ D R˛ f1 .x/ C R˛ f2 .x/; R

x 2 D; f 2 C01 .D/:

.4:4:12/

Here f1 is the extension of f by setting f1 .x/ D 0; x 2 Rd n D: f2 is defined by f2 .x/ D 0 for x 2 D and f2 .x 0 ; xd / D f .x 0 ; xd / for xd  0. It is clear b˛ f 2 H 1 .D/; .˛  .1=2//R b˛ f .x/ D f .x/; x 2 D, and from this that R b limxd #0 .@=@xd /R˛ f .x/ D 0. An integration by parts then gives 1 b˛ f; v/ D .f; v/; b˛ f; v/ C ˛.R D .R 2

v 2 H 1 .D/;

.4:4:13/

where the domain of integration in each term is the half-space D. Equation (4.4.13) b (on L2 .D/) is just ( 1 D; H 1 .D/). implies that the Dirichlet space of M 2 1 1 Since . 2 D; H .D/) is regular on L2 .D/, each function u 2 H 1 .D/ admits a modification e u on D which is quasi continuous on D, or, equivalently, finely b Then we continuous q.e. on D with respect to the reflecting Brownian motion M. can see in the same way as in Example 2.3.1 that u D 0 q.e. on @Dº: H01 .D/ D ¹u 2 H 1 .D/ W e

.4:4:14/

Compare this with (1.2.16). We further see by Theorem 4.3.1 that u 2 H 1 .D/ is ˛ e u. ˛-harmonic if and only if e u D H@D

4.5 Continuity, killing, and jumps of sample paths Just as in the preceding two sections we consider the m-symmetric Hunt process M on X whose Dirichlet space .E; F / on L2 .XI m/ is regular. Recall that the Dirichlet space .E; F / is said to possess the local property if E.u; v/ vanishes whenever u; v 2 F have disjoint compact supports.

179

4.5 Continuity, killing, and jumps of sample paths

Lemma 4.5.1. The following two conditions are equivalent to each other: (i) .E; F / possesses the local property. (ii) For each relatively compact open set A and ˛ > 0, the hitting distribution HX˛nA .x;  / is concentrated on the boundary @A for q.e. x 2 A. Proof. Denote by P the projection operator on the space u D 0 q.e. on X n Aº FA D ¹u 2 F W e in the Hilbert space .F ; E˛ /. By virtue of Theorem 4.3.1, we have for non-negative u 2 F that u is a quasi continuous version of Pu: e u  HX˛nAe

.4:5:1/

Suppose that E possesses the local property. Take a non-negative u 2 F \ C0 .X/ such that suppŒu  X n A. Since Pv D v and suppŒv  A for any v 2 FA , we get E˛ .Pu; v/ D E˛ .u; v/ D 0; 8v 2 FA , which together with (4.5.1) implies HX˛nAe u.x/ D 0 q.e. on A. From this and Lemma 1.4.2, we conclude that HX˛nA .x; X n @A/ D 0 q.e. x 2 A. Conversely, assume condition (ii) and take u; v 2 F with disjoint compact supports. Without loss of generality, we may suppose that u is non-negative. Let A be a relatively compact open set such that suppŒv  A; suppŒu  X n A. Then E.u; v/ D E˛ .u; v/ D E˛ .Pu; v/ which vanishes because Pu D 0 in view of (4.5.1) and the assumption (ii). We call a Hunt process on X a diffusion if Px .X t is continuous in t 2 Œ0; // D 1

.4:5:2/

for every x 2 X. In the paragraph preceding to Theorem 4.2.7, we have defined the equivalence of two m-symmetric Hunt processes on X. Theorem 4.5.1. The following conditions are equivalent to each other: (i) .E; F / possesses the local property. (ii) M is of continuous sample paths for q.e. starting point, i.e., there exists a properly exceptional set N such that .4:5:2/ holds for every x 2 X n N . (iii) There exists an m-symmetric diffusion on X which is equivalent to M. Proof. Obviously, (ii) implies the second condition of Lemma 4.5.1. Furthermore, (ii) is equivalent to (iii) in view of Theorems A.2.8 and A.2.9. Hence, it only remains to prove the implication (i))(ii).

180

4 Analysis by symmetric Hunt processes

Assume the local property of .E; F / and let O be a countable open base of X. We may assume that each A 2 O is relatively compact. By virtue of Lemma 4.5.1 and Theorem 4.1.1, there exists a properly exceptional set N  X such that HX˛nA .x; X n @A/ D 0; 8x 2 A n N;

.4:5:3/

for every A 2 O; ˛ > 0 being fixed. We notice that the relation (4.5.3) then holds for every ˛ > 0 because of ˇ

HX nA .x; X n @A/ D HX˛nA .x; X n @A/ C .˛  ˇ/ Z  RˇA .x; dy/HX˛nA .y; X n @A/; AnN

x 2 A:

Let us put d D ¹! 2  W 9t 2 .0; .!//; X t  .!/ ¤ X t .!/º. On account of the sample path property (M.6), [ [ ¹! 2  W 0 < PX nA .s !/ < 1; X PXnA .s !/ .s !/ 2 X n @Aº: d D A2O s2QC

For each x 2 X n N , the Px -measure of the event inside the braces equals Ex .PXs .X nA < 1; X XnA 2 X n @A/I Xs 2 A/ .Xs ; X n @A/I Xs 2 A n N / D 0: D Ex .HX0C nA Hence (4.5.2) holds. Having completely characterized the sample continuity up to the life time  by means of the local property of the Dirichlet form, our next task is to study the location X  .2 X / of the sample path right before the life time . When X  2 X with positive probability, we say that the killing takes place inside X. In the following we occasionally denote the integral of a function v with a positive measure  by h; vi or hv; i: Z h; vi D hv; i D v.x/.dx/: .4:5:4/ X

Lemma 4.5.2. (i) There exists a unique positive Radon measure k on X charging no exceptional set such that for any u 2 Fe Z 1 lim u.x/2 .1  p t 1.x//m.dx/ D he u2 ; ki; .4:5:5/ t #0 t X e u being any quasi continuous version of u.

4.5 Continuity, killing, and jumps of sample paths

181

(ii) For any f; h 2 B C .X/, and t > 0, Z Ehm .f .X  /I   t/ D

0

t

hf k; ps hids:

.4:5:6/

(iii) 1K  k is of finite energy integral for each compact K. For ˛ > 0 and f 2 C0C .X/, the function Ex .e ˛ f .X  // is a quasi continuous version of the potential U˛ .f  k/. Proof. We first note that the formula Z 1 .u.x/  u.y//2 p t .x; dy/m.dx/ 2t X X Z 1 C u.x/2 .1  p t 1.x//m.dx/ " E.u; u/; t X .4:5:7/ for t # 0, holds for any Borel function u 2 Fe . In fact, the left-hand side equals the approximating form E .t / .u; u/ when u 2 F . (4.5.7) then extends to any u 2 Fe as in Theorem 1.5.2 (ii) where the resolvent rather than the semigroup was used. Hence we can subtract a sequence tn # 0 such that the measure .1=tn /.1  p tn 1.x//m.dx/ converges vaguely to a positive Radon measure k on X and .4:5:8/ he u2 ; ki  E.u; u/; u 2 F \ C0 .X/: In particular, 1K  k is of finite energy integral for any compact set K and, hence, k is smooth (see §2.2). k charges no exceptional set on account of Theorem 4.2.1. Using Theorem 2.1.7, we easily see that the inequality (4.5.8) holds for any u 2 Fe . We can then show that the relation (4.5.5) holds for any u 2 Fe as long as the limit on the left-hand side is taken along the above sequence tn . Indeed, p choosing for any " > 0 a function v 2 F \ C0 .X/ such that E.u  v; u  v/ < ", it follows from (4.5.7) and (4.5.8) that ˇs ˇ q ˇ 1 ˇ 2 2 ˇ ˇ hu ; .1  p 1/  mi he u ; ki t n ˇ ˇ t n ˇ ˇs q ˇ ˇ 1 2 2 ˇ hv ; .1  p tn 1/  mi  hv ; kiˇˇ C 2" ˇ tn which is not greater then 3" for sufficiently small tn . Next let us show the relation (4.5.6) for the measure k determined above. We O hO 2 C C .X/. Notice that h 2 first do this for f 2 C0C .X/ and h D ˛R˛ h; 0

182

4 Analysis by symmetric Hunt processes

D.A/ \ L1 .XI m/; A being the infinitesimal generator of the L2 -semigroup T t determined by p t . We then have Ehm .f .X  /I   t/ D lim Ehm

=tn   ŒtX

n!1

f .X.k1/tn /  1¹.k1/tn 0 such that kps h  hkL2  2skAhkL2 for any s < ı. For tn < ı ˇ ˇ ŒtX Z ˇ ˇ =tn  1 t ˇ .p.k1/tn h; f .1  p tn 1//  .ps h; f .1  p tn 1//ds ˇˇ ˇ tn 0 kD1



ŒtX =tn  kD1

C

1 tn

1 tn Z

Z

k tn

.k1/tn

j.p.k1/tn h  ps h; f .1  p tn 1//jds

t

t Œt =tn 

j.ps h; f .1  p tn 1//jds

 2t kAhkL2  kf .1  pı 1/kL2 C khkL2 kf .1  pı 1/kL2 ; which decreases to zero as ı # 0. Therefore, Ehm .f .X  /I   t / D limn!1 t1n  Rt Rt 0 .ps h; f .1  p tn 1//ds D 0 hps h  f; kids proving (4.5.6) for the present f and h. Letting ˛ tend to infinity and then using the monotone lemma, we know that (4.5.6) holds for any f; h 2 B C .X/. Taking the Laplace transform of both sides of (4.5.6), we readily see that the function w.x/ D Ex .e ˛ f .X  // is a version of the potential U˛ .f  k/ for f 2 C0C .X/. Since w is ˛-excessive, it is quasi continuous by Lemma 4.2.2, completing the proof of the third statement (iii). In particular, the Radon measure k is uniquely determined by the process M independently of the choice of ¹tn º. Hence, the proof of (i) is also complete. We now identify the measure k in the above theorem with the killing measure (associated with the form E) appearing in the Beurling–Deny formula.

4.5 Continuity, killing, and jumps of sample paths

183

Lemma 4.5.3. The measure k in Lemma 4:5:2 coincides with the killing measure in Theorem 3:2:1. Proof. Denote by e k the killing measure in the statement of Theorem 3.2.1. In accordance with the construction of e k there, consider a sequence of relatively comk was constructed in pact open sets Gl increasing to X. The difficulty lies in that e two steps and, in fact, e k is a vague limit of kl determined by (3.2.9). However, a similar reasoning to the first part of the proof of Lemma 4.5.2 applies yielding that lim he u2 ; kl i D he u2 ; e ki .4:5:9/ l!1

for any u 2 F with compact support. Let E .l/ and M.l/ be the parts on the open set Gl of the Dirichlet form E and the Hunt process M respectively. By virtue of Theorem 4.4.3, we can apply Lemma 4.5.2 to M.l/ and E .l/ getting a positive Radon measure kl0 on Gl such that for any u 2 FGl 1 lim t #0 t

Z 2

Gl

u.x/ .1 

.l/ p t 1.x//m.dx/

Z D

Gl

e u.x/2 kl0 .dx/

.4:5:10/

and for any f; h 2 C0C .X/ and sufficiently large l Z Ehm .f .Xl  /I l  t/ D

t 0

hf  kl0 ; ps.l/ hids;

.4:5:11/

.l/

where l D X nGl ^  and p t is the transition function of M.l/ . Comparing (4.5.10) with (3.2.9), we have kl0 .E/  kl .E/; E  Gl . Hence, the right-hand side of (4.5.11) is not less than Z 0

Z

t

kl ; ps.l/ hids

hf 



t 0

hf  kl ; ps.l0 / hids;

l > l0 :

Since the left-hand side of (4.5.11) tends to Ehm .f .X  /I   t /, we get from (4.5.6) and (4.5.9) the inequality 1 t

Z 0

t

1 hf  k; ps hids  t

Z 0

t

hf  e k; ps.l0 / hids:

k; hi. The After letting l0 ! 1 and then t # 0, we arrive at hf  k; hi  hf  e converse inequality is obvious and, consequently, k D e k.

184

4 Analysis by symmetric Hunt processes

Finally, we are concerned with the jumping measure J appearing in the Beurling–Deny formula: J is the unique positive Radon measure on X  X n d such that for any u; v 2 F \ C0 .X/ with disjoint support, Z 2 u.x/v.y/J.dxdy/ D E.u; v/: .4:5:12/ Lemma 4.5.4. (i) J charges no part of X  X n d whose projection on the factor X is exceptional. (ii) The formula (3.2.1) extends to any u; v 2 Fe : Z E.u; v/ D E .c/ .u; v/ C .e u.x/  e u.y//.e v .x/  e v .y//J.dxdy/ X X nd

Z C

X

e u.x/e v .x/k.dx/;

(4.5.13)

where e u and e v denote quasi continuous versions and E .c/ is a symmetric form .c/ with DŒE  D Fe satisfying for any u; v 2 Fe of compact support E .c/ .u; v/ D 0 if v is constant on a neighbourhood of suppŒu: .4:5:14/ Proof. Consider a relatively compact open set G and a non-negative v 2 F \ C0 .X/ with R suppŒv  X n G. Then (4.5.12) implies that the measure Jv .dx/ D 2  1G .x/ v.y/J.dxdy/ is of finite energy integral with respect to the part EG of the Dirichlet form E on the set G. Z p ju.x/jJv .dx/  C E.u; u/; 8u 2 FG \ C0 .G/: .4:5:15/ G

Hence Jv charges no set of zero capacity with respect to EG and, consequently, no exceptional set in view of Theorem 4.4.3. This together with Lemma 4.5.2 implies that the integrals in (4.5.13) are independent of the choice of the versions e u and e v . Let us define E .c/ by the equation (4.5.13). By virtue of Theorem 2.1.7 and Theorem 3.2.1, E .c/ then becomes a symmetric form. Its property (3.2.2) extends to (4.5.14) by working with the perturbed form .E g ; Feg / of Lemma 1.6.7 as in the proof of Theorem 3.1.2 and utilizing the expression v D .v  v0 / C v0 with v0 2 F \ C0 .X/ such that v0 D v D constant on suppŒu. We can derive the following identity from formula (4.5.12) and Theorem 4.4.1. For G; v and Jv as in the above proof v  H˛X nG v D U˛G Jv ;

˛ > 0;

.4:5:16/

4.5 Continuity, killing, and jumps of sample paths

185

where U˛G Jv is the ˛-potential of Jv with respect to the form EG . In particular, for any non-negative Borel h vanishing outside G Z ˛G v.XG // D R˛G h.x/v.y/J.dxdy/; .4:5:17/ Ehm .e G being the first leaving time from G W G D X nG ^ &. This formula is strengthened in the next lemma. Lemma 4.5.5. Let G be a relatively compact open set. (i) For any h; f; g 2 BbC .X/ such that suppŒh  G; suppŒf   G and suppŒg  X n G, Ehm .f .XG  /g.XG /I G  t/ Z t Z D2 psG h.x/f .x/g.y/J.dxdy/ ds:

.4:5:18/

0

(ii) The function Ex .e ˛G f .XG  /g.XG // on G is a quasi continuous version of the potential U˛G .fJg / for ˛ > 0 and f; g 2 C0C .X/ with suppŒf   G, suppŒg  X n G. Proof. The proof proceeds similar to the proof of Lemma 4.5.2. In fact, when h satisfies the condition in the proof of Lemma 4.5.2, we see that the left-hand side of Rt PŒt =ı G h; f pı g/ D limı#0 1ı 0 .psG h; f pı g/ds, (4.5.18) equals limı#0 kD1 .p.k1/ı 2 2 2 2 2 where we use the estimate kf  pı gkL 2  .f ; pı g /!ı#0 .f ; g / D 0. The last limit is equal to the right-hand side of (4.5.18) because the preceding lemma implies Z 1 2 e u.x/e v .y/J.dxdy/ D E.u; v/ D lim .u; ps v/ s#0 s for any u; v 2 F with compact disjoint support. Summing up Lemmata 4.5.2–4.5.5, we can formulate a refinement of Theorem 3.2.1 as follows: Theorem 4.5.2. The Dirichlet form E.u; v/; u; v 2 Fe , admits a unique representation .4:5:13/ with E .c/ ; J and k possessing the following properties: The local part E .c/ is a symmetric form with DŒE .c/  D Fe and has the strong local property .4:5:14/.

186

4 Analysis by symmetric Hunt processes

The jumping measure J is a unique positive Radon measure on X  X n d satisfying .4:5:12/. J indicates the jumps of the sample paths in the sense of the identity .4:5:18/. The killing measure k is a unique positive Radon measure satisfying .4:5:5/. k indicates the killing of the sample paths inside X in the sense of the identity .4:5:6/. Combining this theorem with Lemma 4.5.2 (iii) and Theorem 4.5.1, we get Theorem 4.5.3. The following conditions are equivalent to each other: (i) E possesses the strong local property .4:5:14/. (ii) E possesses the local property and k D 0. (iii) M is of continuous sample paths and with no killing inside for q.e. starting points, i.e., there exists a properly exceptional set N such that .4:5:2/ and Px .X  2 X;  < 1/ D 0

.4:5:19/

hold for every x 2 X n N . (iv) M admits an equivalent m-symmetric diffusion on X satisfying .4:5:19/ for every x 2 X. Let us define the resurrected form E res by ´ R E res .u; v/ D E.u; v/  X e u.x/e v .x/k.dx/ DŒE res  D Fe : By (4.5.13), E res is a symmetric form expressible as Z res .c/ u.x/  e u.y//.e v .x/  e v .y//J.dxdy/; E .u; v/ D E .u; v/ C .e

.4:5:20/

u; v 2 Fe :

.4:5:21/ On the other hand, the relation (4.5.7) implies that every normal contraction operates on E res and Z 1 1 2 res e u.x/2 k.dx/  E.u; u/ lim Em ..u.X t /  u.X0 // / D E .u; u/ C 2 X t #0 2t .4:5:22/ 7 for any Borel u 2 Fe . This formula plays an important role in the next chapter. E is called conservative if the associated semigroup T t on L2 .XI m/ satisfies T t 1 D 1; 8t > 0. 7 Recall

our convention that u./ D 0.

4.5 Continuity, killing, and jumps of sample paths

187

Exercise 4.5.1. Prove the equivalence of the next conditions. (i) E is conservative. (ii) Px . < 1/ D 0 for q.e. x 2 X:

(4.5.23)

(iii) M admits an equivalent m-symmetric Hunt process on X which is conservative in the sense that (4.5.23) holds for every x 2 X. If M satisfies the extra condition of absolute continuity (4.2.9), then any ˛-excessive function vanishing m-a.e. is identically zero .˛  0/. This combined with Exercise 4.2.2 enables us to strengthen several assertions obtained so far into “everywhere” assertions as follows. Theorem 4.5.4. Assume the absolute continuity condition for M. (i) For ˛ > 0 and f 2 C0C .X/ Ex .e ˛ f .X  // D R˛ .f  k/.x/;

x 2 X;

.4:5:24/

where the right side is defined by .4:2:14/. (ii) .E; F / possesses the local property if and only if M is a diffusion. (iii) .E; F / possesses the strong local property if and only if M is a diffusion and Px .X  2 X;  < 1/ D 0;

8x 2 X:

.4:5:25/

(iv) .E; F / is conservative if and only if M is conservative. In particular, the d -dimensional Brownian motion formulated in Example 4.1.1 and stated in Example 4.2.1 is a conservative diffusion on Rd in virtue of this theorem. Example 4.5.1 (one-dimensional diffusion). Let I D .r1 ; r2 / be a one-dimensional interval and let m and k be positive Radon measures on I with suppŒm D I . We consider an m-symmetric Hunt process M on I whose Dirichlet space on L2 .XI m/ is given by .F 0 ; E/ of Example 1.2.2. As we saw in Example 2.1.2, no non-empty subset of I is of capacity zero. Accordingly any non-empty subset of I is neither exceptional nor polar with respect to the process M (Theorem 4.2.1). In particular M satisfies the absolute continuity condition (4.2.9) in virtue of Theorem 4.2.4. From the expression (2.1.19) of the 1-equilibrium potential p.x/ of a one-point set ¹yº, we know that p.x/ is strictly positive. Hence Theorem 4.2.5 implies that Px .y < 1/ > 0;

8x; y 2 I;

.4:5:26/

namely, any two points have the communication with respect to the process M.

188

4 Analysis by symmetric Hunt processes

Since E possesses the local property, we see by either Theorem 4.5.1 or Theorem 4.5.4 that M is a diffusion process on I with possible killing inside I indicated by the killing measure k. When k D 0 and m D Lebesgue measure on I , then .F 0 ; E/ D .H01 .I /; 12 D/ and, consequently, M is the absorbing Brownian motion on the interval I in view of Example 4.4.1. Finally we give two examples of multidimensional symmetric diffusions satisfying the absolute continuity condition (4.2.9). Example 4.5.2 (a diffusion associated with a second order differential operator of divergence form). The Dirichlet form associated with a second order differential P operator di;j D1 @x@ .aij .x/ @x@ / has been considered already in §1.6, §3.1 and i

j

§3.3. We consider a family ¹aij .x/º1i;j d of Borel functions on Rd satisfying aij .x/ D aj i .x/; 1  i; j  d , and d d d X X 1X 2 i  aij .x/i j   i2 ;  iD1

i;j D1

8 2 Rd ; 8x 2 Rd

.4:5:27/

F D H 1 .Rd /;

.4:5:28/

iD1

for some constant  > 1. Then E.u; v/ D

d Z X i;j D1

Rd

aij .x/

@u @v dx; @xi @xj

is a regular Dirichlet form on the space L2 .Rd / based on the Lebesgue measure. By invoking the well known works on PDE,8 one can see that the above Dirichlet form admits a Markovian resolvent ¹R˛ º˛>0 on L2 .Rd / which further satisfies the following properties:9 R˛ .L2 .Rd / \ Lp .Rd //  C1 .Rd /; R˛ .C1 .Rd //

p > d;

is a dense subset of C1 .Rd /:

(4.5.29) (4.5.30)

By (4.5.30) and the Hille–Yosida theorem,10 ¹R˛ º˛>0 produces a strongly continuous Markovian semigroup on C1 .Rd /, and accordingly E admits a Feller transition function p t and an associated Hunt process M on Rd (Theorem A.2.2). Since R˛ .x;  / is absolutely continuous with respect to the Lebesgue measure in view of (4.5.29), p t .x;  / has the same property by Theorem 4.2.4. 8 Cf.

G. Stampacchia [2]. H. Kunita [2], M. Tomisaki [2]. 10 Cf. K. Yosida [1]. 9 Cf.

4.6

189

Quasi notions, fine notions and global properties

Obviously E has the local property. As we shall see in Example 5.7.1, E is also conservative. Hence Theorem 4.5.4 implies that M is a conservative diffusion on Rd . Example 4.5.3 (reflecting Brownian motion on a Lipschitz domain). Let D be a bounded domain of Rd which is assumed to be Lipschitz in the sense that each boundary point x admits a neighbourhood U such that D \ U is expressible as in Example 1.6.1by a Lipschitz continuous function f . According to Example 1.6.1, the form E D 12 D; H 1 .D/ is then a regular Dirichlet form on L2 .D/. It is further known thatE admits  a transition function p t on D which is strong Feller in the sense that p t Bb .D/  C.D/ and lim t #0 p t u.x/ D u.x/, 8x 2 D, 8u 2 C.D/.11 There exists a Hunt process M on D with the absolutely continuous transition function p t (Theorem A.2.2). Clearly E possesses the local property. It is also conservative (see Example 1.6.1 and Lemma 1.6.5). Hence M is a conservative diffusion on D by virtue of Theorem 4.5.4. M is called the reflecting Brownian motion on D.

4.6 Quasi notions, fine notions and global properties We maintain the preceding setting: M is an m-symmetric Hunt process whose Dirichlet form E on L2 .XI m/ is regular. In §4.2, we have studied the relation of quasi continuity and q.e. fine continuity of functions. We can now complete this relationship. For two subsets E1 ; E2 of X, we write E1  E2

q.e.

.4:6:1/

if E1 n E2 is exceptional or, equivalently, Cap.E1 n E2 / D 0 (see Theorem 4.2.1). Thus E1 D E2 q.e. iff the symmetric difference E1 E2 is exceptional. In this case, E1 and E2 are said to be q.e. equivalent. Obviously the notion of quasi open sets and quasi closed sets introduced in §2.1 has the class property in the sense of q.e. equivalence. A set E is called q.e. finely open if E is q.e. equivalent to a nearly Borel finely open set. Theorem 4.6.1. (i) A set E is quasi open if and only if E is q.e. finely open. (ii) A numerical function u is quasi continuous if and only if u is finite q.e. and u is q.e. finely continuous. 11 Cf.

R. F. Bass and P. Hsu [2].

190

4 Analysis by symmetric Hunt processes

Proof. (i) If E is quasi open, there is a nest ¹Fk º such that E \ Fk is relatively for every k. Let Ak D X nFk and choose a properly exceptional set N open on FkT containing 1 kD1 Ak such that (4.2.5) holds. Then E n N is Borel and finely open. e Conversely suppose that E is q.e. equivalent to a nearly Borel finely open set E. 2 Take a strictly positive bounded Borel function f 2 L .XI m/ and let Z  Xne E s e f .Xs /ds ; x 2 X: .4:6:2/ v.x/ D Ex 0

Then v is quasi continuous by Theorem 4.4.1 and E D v 1 .0; 1/ q.e. Hence E is quasi open. (ii) Suppose u is finite q.e. and q.e. finely continuous, then, for any open I  R, u1 .I / is q.e. finely open and consequently quasi open by (i). Hence u is quasi continuous. The converse was proven in Theorem 4.2.2. The second subject of this section is about quasi supports of measures. We prepare a simple lemma. Lemma 4.6.1. A set F is quasi closed if and only if there exists a non-negative quasi continuous function u in F with F D u1 .¹0º/ q.e. Proof. The “if” part is evident. Conversely, given a quasi closed set F , we can e being replaced take as a required function u the function v of (4.6.2) with X n E e q.e. equivalent to F in accordance with by a nearly Borel finely closed set F Theorem 4.6.1. Let us consider a positive Borel measure  on X charging no set of zero cae is said to be a quasi support of  if the next two conditions are pacity. A set F satisfied: e is quasi closed and .X n F e / D 0; F e  FL q.e. if FL is another set with property (4.6.3), then F

(4.6.3) (4.6.4)

e of  is unique up to the q.e. equivalence. Let F D suppΠThe quasi support F be the topological support of . Since any closed set is quasi closed, we have that e  F q.e., and by deleting a set of zero capacity from F e if necessary, we can F e  F. always assume that F Theorem 4.6.2. The following three conditions are equivalent to each other for any Borel measure  on X charging no set of zero capacity and for any quasi closed set F  X:

4.6

Quasi notions, fine notions and global properties

191

(i) F is a quasi support of . (ii) u D 0 -a.e. on X if and only if u D 0 q.e. on F for any quasi continuous u2F. (iii) Condition (ii) holds for any quasi continuous function u. Proof. (ii) ) (i): We set ² ³ Z N D u 2 F W je ujd D 0 ;

(4.6.5)

FF c D ¹u 2 F W e u D 0 q.e. on F º

(4.6.6)

X

and assume that N D FF c . Take a function u of Lemma 4.6.1. Then u 2 FF c and hence u 2 N , which means .X n F / D 0. Consider another quasi closed set F1 with .X n F1 / D 0 and take again a function u1 satisfying the condition of Lemma 4.6.1 for F1 . Then u1 2 N and hence u1 2 FF c , which implies that F  F1 q.e., proving that F is a quasi support of . (i) ) (iii): Suppose F is a quasi support of , then the “if” part of condition (ii) is clearly satisfied for any Borel function u. If u is quasi continuous and u D 0 -a.e., then the set FL D ¹u D 0º has the property (4.6.3) and hence F  FL q.e. and u D 0 q.e. on F . The implication (iii) ) (ii) is trivial. In view of Lemma 2.1.4, we get Corollary 4.6.1. If  D f  m for some measurable function f strictly positive m-a.e. on X, then  has the full quasi support X. Exercise 4.6.1. Suppose that F is a Borel quasi closed set,  is a positive Radon measure charging no set of zero capacity, .X n F / D 0 and the 1-order hitting distribution HF1 .x;  / is absolutely continuous with respect to  for m-a.e. x 2 X. Show that F is a quasi support of . Recall the class S of smooth measures introduced in §2.2. Each  2 S charges no set of zero capacity. Theorem 4.6.3. Any  2 S admits a quasi support. Proof. For any  2 S , the space N defined by (4.6.5) is a closed subspace of the closed sets separable Hilbert S space .E1 ; F / as 1Fn   2 S0 for some increasing e is continuously embedded into L1 .XI 1F  / Fn with .X n n Fn / D 0 and F n

192

4 Analysis by symmetric Hunt processes

for each n by Theorem 2.2.2. Choose a countable dense subcollection ¹uk º of N and let 1 X je uk j.x/ 1 F D v .¹0º/ for v.x/ D 2k : kuk kE1 kD1

Since the series defining v is E1 -convergent, v is quasi continuous and v 2 N . Hence F is quasi closed and .X n F / D 0. Accordingly we arrive at the identity N D FF c for FF c defined by (4.6.6) for this set F . By virtue of Theorem 4.6.2, we can conclude that F is a quasi support of . In §5.1, we shall identify the quasi support of  2 S with the support of the positive continuous additive functional A associated with . Recall Theorem 4.4.2 stating that the part EG of E on an open set G  X is a regular Dirichlet form on L2 .GI m/ and it is associated with the part MG of M on G which is a Hunt process. Exercise 4.6.2. Consider  2 S with the quasi support F: Let G be a open subset of X: ˇ (i) Show that G \ F is a quasi support of ˇG relative to the Dirichlet form EG : (ii) Suppose a quasi continuous function u on G vanishes -a.e. on G: Show that u D 0 q.e. on G \ F: We turn to the study of the ergodic decomposition of the state space X into M-invariant sets. The quasi notions work well in this respect. Recall that an mmeasurable set A is said to be T t -invariant if T t .1A f / D 1A T t f m-a.e. for any f 2 L2 .XI m/ and t > 0 (§1.6). Given two m-measurable sets A1 and A2 , we say that A1 is a modification of A2 if m.A1 A2 / D 0. Lemma 4.6.2. Any T t -invariant set A admits a modification which is simultaneously quasi open and quasi closed. Proof. If A is T t -invariant, then 1A  u 2 F for any u 2 F by Theorem 1.6.1. Therefore 1A possesses a quasi continuous modification, say ', on each relatively compact open set and hence on the whole space X. Since ' 2 D ' q.e., ' D 0 or 1 q.e., and there is a nearly Borel set e A with ' D 1 e A q.e. A set B  X has been called M-invariant if B is nearly Borel and Px .X t 2 B ; X t  2 B ; 8t  0/ D 1;

8x 2 B:

Obviously any M-invariant set is T t -invariant. The Hunt process M is said to be a diffusion if X t is continuous in t 2 Œ0; / Px -a.s. for any x 2 X. M admits an

4.6

Quasi notions, fine notions and global properties

193

equivalent diffusion if and only if the Dirichlet form E possesses the local property (Theorem 4.5.1). Lemma 4.6.3. Suppose that a set B is simultaneously quasi open and quasi closed. We further assume that one of the following two conditions is satisfied: B is T t -invariant;

(4.6.7)

M is a diffusion:

(4.6.8)

Then there is a properly exceptional set N such that both B n N and .X n B/ n N are M-invariant. Proof. Let B be quasi open and quasi closed. Then 1B is quasi continuous and by virtue of Theorem 4.2.2 there is a properly exceptional set N such that 1B is nearly Borel on X n N and, for any x 2 X n N ,   Px 1B .X t / is right continuous and lim 1B .X t 0 / D 1B .X t  /; 8t 2 Œ0; / D 1; t 0 "t



(4.6.9)



Px lim 1B .X t 0 / D 1B .X  /; X  2 X D Px .X  2 X/: t 0 "

(4.6.10)

Assume first condition (4.6.8). Then 1B .X t / is continuous in t 2 Œ0; / Px -a.s. for x 2 X n N . Consequently Px .X t 2 B; 8t 2 Œ0; // D 1;

8x 2 B n N;

Px .X  2 B/ D Px .X  2 X/;

8x 2 B n N

and the same equations hold with B being replaced by X n B. Therefore both B n N and .X n B/ n N are M-invariant. Next assume (4.6.7) instead of (4.6.8). In view of Lemmas 4.2.3 and 2.1.4, p t .1B u/ D 1B  p t u q.e.;

8t > 0; 8u 2 C0 .X/:

By taking un 2 C0C .X/ increasing to 1, we get p t 1B D 1B  p t 1 q.e. for each t > 0, which in conjunction with (4.6.9) and (4.6.10) implies that B n N and .X n B/ n N are M-invariant for an appropriate properly exceptional set N . Here are two immediate consequences of the preceding two lemmas. Corollary 4.6.2. The following conditions are equivalent to each other for an mmeasurable set B:

194

4 Analysis by symmetric Hunt processes

(i) B is T t -invariant, (ii) there is a nearly Borel modification B1 (resp. B2 ) of B (resp. X n B) such that both B1 and B2 are M-invariant, X D B1 C B2 C N

.disjoint sum/

and m.N / D 0. Exercise 4.6.3. Suppose ¹p t º is strongly Feller. Show that, if the state space X is connected, then M is irreducible. Corollary 4.6.3. Suppose that the Dirichlet form E has the local property. Then the following conditions are equivalent to each other for an m-measurable set B: (i) B is T t -invariant, (ii) B admits a modification which is quasi open and quasi closed simultaneously. The Dirichlet form E or the associated semigroup T t is said to be irreducible if any T t -invariant m-measurable set B is m-trivial in the sense that either B or B c is m-negligible. We may say by Corollary 4.6.3 and Lemma 2.1.7 that, under the condition of the local property of E, E is irreducible if and only if the space X is quasi-connected in the sense that, if B is simultaneously quasi open and quasi closed, then either B or B c is of zero capacity. Theorem 4.6.4 (comparison of irreducibility). Consider two regular Dirichlet forms E .i / on L2 .XI m.i / /; i D 1; 2. We suppose that E .1/ possesses the local property. We further assume that m.2/ is absolutely continuous with respect to m.1/ ;

(4.6.11)

E .2/ -quasi continuity implies E .1/ -quasi continuity:

(4.6.12)

Then the irreducibility of E .1/ implies the same property of E .2/ . Proof. By Theorem 7.2.2, we may assume that a diffusion process is associated .i / with E .1/ . We denote by T t the semigroup on L2 .XI m.i / / associated with E .i / , .2/ i D 1; 2. Let B be a T t -invariant m.2/ -measurable subset of X. Then 1 e B is .2/ .2/ e with m .B B/ e D 0 by virtue of E -quasi continuous for some Borel set B .1/ Lemma 4.6.2. By assumption (4.6.12), 1 e -quasi continuous and hence B is E .1/ .1/ e is m.1/ -trivial and T t -invariant by Corollary 4.6.3. If E is irreducible, then B .2/ consequently B is m -trivial by assumption (4.6.11) proving the irreducibility of E .2/ .

4.6

Quasi notions, fine notions and global properties .1/

.2/

is locally dominated by E1

The condition (4.6.12) is fulfilled if E1 formulated below.

195 as is

Corollary 4.6.4. Consider two Dirichlet forms E .i / on L2 .XI m.i / / possessing a common special standard core C. We suppose that E .1/ has the local property. We further assume that there are open sets Gn increasing to X and satisfying the next conditions: d m.2/ D d m.1/

with ess inf .x/ > 0; n D 1; 2; : : : ; x2Gn

E .2/ .u; u/  n E .1/ .u; u/;

8u 2 CGn

(4.6.13) (4.6.14)

where n is a positive constant and CGn is defined by (2.3.12) for each n. Then the irreducibility of E .1/ implies the same property of E .2/ . Proof. A function is quasi continuous on X if and only if it is so on each Gn . Therefore (4.6.12) follows from (4.6.13) and (4.6.14) by virtue of Theorem 4.4.3.

Example 4.6.1. Let X be a domain D of Rd and m be an everywhere dense positive Radon measure on D. We consider a Dirichlet form E on L2 .DI m/ possessing C01 .D/ as its core and admitting the expression (1.2.1) on C01 .D/. Denote by V the Lebesgue measure. We assume the following coerciveness condition: d X i;j D1

1 i j ij .K/  K jj2 V .K/ 2

for any compact set K  D and  2 Rd , where K is a positive constant. Then 1 E.u; u/  EN D.u; u/; 2

u 2 C01 .E/

for any relatively compact open subset E of D. We further assume that d m D dV

with inf .x/ > 0 K

for any compact K  D. Since the Sobolev space . 12 D; H01 .D// of Example 1.2.3 is irreducible, E is also irreducible in virtue of Corollary 4.6.4.

196

4 Analysis by symmetric Hunt processes

Returning to the previous setting that M is an m-symmetric Hunt process on X whose Dirichlet form E on L2 .XI m/ is regular, we consider a nearly Borel set B  X. We are interested in a specific ergodic decomposition of X related to the hitting probability of B; we put Y D ¹x 2 X W Px .B < 1/ > 0º:

.4:6:15/

Lemma 4.6.4. There is a properly exceptional set N such that both Y n N and .X n Y / n N are M-invariant. Proof. It suffices to prove that Px .Y < 1/ D 0;

8x 2 X n Y:

.4:6:16/

Indeed (4.6.16) implies that Y is T t -invariant by Lemma 1.6.1 and that X n Y is finely open. Since Y is also finely open, we see by Theorem 4.6.1 that Y is simultaneously quasi open and quasi closed and hence Lemma 4.6.3 is applicable. To prove (4.6.16), we introduce the finely closed set ² ³ 1 : Yn D x 2 X W Px .B < 1/  n Then, for x 2 X n Y , Px .Yn < 1/ D Px .B D 1; Yn < 1/ D Ex .PXY .B D 1/ I B D 1; Yn < 1/ n   1 Px .Yn < 1/;  1 n which means Px .Yn < 1/ D 0. We arrive at (4.6.16) by letting n ! 1. e the M-invariant set Y n N and by G the fine interior of the Let us denote by Y e set Y n B. Accordingly e n B W Px .B > 0/ D 1º: G D ¹x 2 Y

.4:6:17/

Owing to Theorem 4.4.2, we see that the part EG of the Dirichlet form E on the set G is a Dirichlet form on L2 .GI m/ and associated with the part MG of M on e n B, the set G. If we denote by R˛G the resolvent of MG , then, for any x 2 Y   Z B ^ G t R1 1.x/ D Ex e dt  1  Ex .e  B / < 1: 0

Therefore we get the next lemma by virtue of Lemma 1.6.5.

4.6

Quasi notions, fine notions and global properties

197

Lemma 4.6.5. The part EG of E on the set G is a transient Dirichlet form on L2 .GI m/. By making use of the preceding two lemmas, we can prove a useful property of the extended Dirichlet space Fe for the Dirichlet form E. We do not assume the transience of E. Nevertheless we know from Theorem 2.1.7 that any u 2 Fe admits a quasi continuous modification, which we shall denote by e u. The next lemma is a key step to extend Theorem 4.3.2 to the present general case. uj.x/ is finite m-a.e. Lemma 4.6.6. For any u 2 Fe , HB je Proof. We may assume by Corollary 1.6.3 that u 2 Fe is non-negative. Since HBe u.x/ D e u.x/ < 1 q.e. x 2 B [ B r by Theorem 4.1.3 and HBe u.x/ D 0, 8x 2 X n Y , it suffices to show that u.x/ < 1 HBe

m-a.e. on G:

.4:6:18/

Since EG is transient, it admits a reference functionR g on G: g is a strictly positive bounded m-integrable function on G such that G g  RRG gd m  1. By replacing g with g=.u _ 1/ if necessary, we may suppose that G gud m < 1. Recall that the domain of EG is given by FG D ¹v 2 F W e v D 0 q.e. on X n Gº: We let un D u ^ n. For a fixed n, we can take on account of Corollary 1.6.3 and Theorem 2.1.7 a sequence of functions vk 2 F such that 0  vk  n, ¹vk º is E-Cauchy and limk!1 e vk D e un q.e. In view of Theorem 1.6.1 and Theo.v  HB˛e v k / 2 FG and rem 4.3.1, we have that 1 e Y k Z

G

˛ gjvk  HB˛e v k jd m D E˛ .R˛G g; 1 e Y jvk  HB vk j/



1=2

Z G

gR˛G gd m

 1=2 E˛ vk  HB˛e v k ; vk  HB˛e vk  E˛ .vk ; vk /1=2 :

By letting ˛ # 0 and k ! 1, we get Z gjun  HBe un jd m  E.un ; un /1=2  E.u; u/1=2 : G

Hence

Z

Z G

g  HBe un d m 

G

g  un d m C E.u; u/1=2

198

4 Analysis by symmetric Hunt processes

and consequently Z

Z G

g  HBe ud m 

G

g  ud m C E.u; u/1=2 < 1;

proving (4.6.18). e e the collection of all quasi continuous functions in Fe . We further Denote by F set u D 0 q.e. on Bº: Fe;X nB D ¹u 2 Fe W e We are now in a position to extend Theorem 4.3.2 from the transient case to the present general case in the following manner. ee Theorem 4.6.5. For any u 2 Fe and any nearly Borel set B  X, HBe u 2 F and E.HBe u; v/ D 0; 8v 2 Fe;X nB . We prove this by a method of perturbation embodied in Lemma 1.6.7. Let us fix a function g in the class K of (1.6.14) for a moment and consider the associated perturbed Dirichlet form .E g ; F / on L2 .XI m/. We also consider the canonical subprocess Mg D .X tg ;  g ; Px / of M relative to the multiplicative functional e C t where Z t g.Xs /ds: .4:6:19/ Ct D 0

According to Theorem A.2.11, Mg is a Hunt process on X with transition function p gt f .x/ D Ex .e C t f .X t //;

x 2 X:

.4:6:20/

Lemma 4.6.7. p gt is an m-symmetric transition function on X whose associated Dirichlet form on L2 .XI m/ coincides with E g . g g Proof. Denote by R˛ f the Laplace transform of p t f :

Z R˛g f .x/ D Ex

1

 e ˛t C t f .X t /dt ;

0

x 2 X:

We have then the well-known formula R˛g f  R˛ f C R˛ .g  R˛g f / D 0:

.4:6:21/

4.6

Quasi notions, fine notions and global properties

199

Rt In fact, using the identity e C t  1 D 0 e Cs g.Xs /ds and the Markov property, we have for a bounded Borel f  Z 1 g ˛t C t C t e e .e  1/f .X t /dt R˛ f .x/R˛ f .x/ D Ex Z D Ex

e 1

e

Z

˛s

Z

e

˛s

0

e

EXs

  D R˛ g  R˛g f .x/;

˛t C t

e

 f .X t /dt

s 1

e 0

1

1

g.Xs /ds

0

Z D Ex

Cs

0

Z D Ex

0

1

˛t C t ıs

Z

e

1

e

 f .X t ı s /dt g.Xs /ds

˛t C t

e







f .X t /dt g.Xs /ds

0

x 2 X:

(4.6.21) implies that, for any bounded Borel f 2 L2 .XI m/,   R˛g f 2 F ; E˛g R˛g f; v D .f; v/; 8v 2 F ;

.4:6:22/

which readily extends to any f 2 L2 .XI m/, as was to be proved. E g is a transient regular Dirichlet form and shares the quasi notion in common   with E because E1g and E1 are equivalent metrics on F . Let Feg ; E g be the extended Dirichlet space of .E g ; F /. It is a Hilbert space and admits an orthogonal decomposition g ˚ HBg ; Feg D Fe;X nB

g Fe;X D ¹u 2 Feg W e u D 0 q.e. on Bº: nB

The orthogonal projection operator P g on HBg is given by P g u.x/ D HBge u.x/ and HBge u.x/ D Ex .exp.C B /e u.X B //; because u.x/ D Ex .e u.X gB // D Ex .e u.X B /I C B < Z/ D Ex .exp.C B /e u.X B // HBge where Z is a relevant exponentially distributed random variable independent of X t (see A.2). The 0-order resolvent of the part Mg;X nB of Mg on X n B is given by   Z B g;X nB C t R f .x/ D Ex e f .X t /dt : 0

Lemma 4.6.8. (i) For any non-negative Borel w HB w.x/  HBg w.x/ D Rg;X nB .g  HB w/.x/ q.e.

.4:6:23/

200

4 Analysis by symmetric Hunt processes

(ii) If w 2 Feg and w is bounded, then HB w e 2 Feg and E.HB w e; v/ D 0; 8v 2 g . Further E.H w e ; H w e /  E.w; w/. Fe;X B B nB Proof. (i) The right-hand side of (4.6.23) equals  Z B  Z 1  C t C t Ex e.X B // dC t e HB w e.X t /dC t D Ex e 1Œ0; B  .t /EX t . w 0

Z

0



0

D Ex

1

e

C t

 1Œ0; B  .t /e w .X B . t !//dC t

Z

D Ex w e.X B /

B

e

C t

 dC t

0

e.X B /.1  e CB //: D Ex . w (ii) Since g is a reference function for .F g ; E g /, we know from Theorem 4.4.1 g that, for any bounded Borel h, Rg;X nB .h  g/ 2 Fe;X and nB E g .Rg;X nB .h  g/; v/ D .h; v/gm ;

g 8v 2 Fe;X : nB

If w 2 Feg and w is bounded, then (i) implies that HB w e 2 Feg and e; v/ D E g .HB w e; v/  .HB w e; v/gm E.HB w D .HB w e; v/gm  .HB w e; v/gm D 0;

g 8v 2 Fe;X : nB

g Since w  HB w e 2 Fe;X , we further obtain nB

e; HB w e/ C E.w  HB w e; w  HB w e /  E.HB w e; HB w e /: E.w; w/ D E.HB w Proof of Theorem 4:6:5. For arbitrarily fixed u 2 Fe and v 2 Fe;X nB , we can g find g 2 K with u; v 2 Feg owing to Lemma 1.6.7. Then v 2 Fe;X . Applying nB g C Lemma 4.6.8 to the truncated functions un D .0 _ u/ ^ n 2 Fe , we get that g HBe uC n 2 Fe and uC E.HBe n ; v/ D 0; C C uC uC kHBe n  HB e m kE  kun  um kE ! 0;

n; m ! 0:

uC uC which is finite m-a.e. owing to Lemma 4.6.6, Since HBe n converges to HB e C u 2 Fe ; kHBe uC uC kE ! 0 and E.HBe uC ; v/ D 0. we conclude that HBe n  HB e The same is true for HBe u . u is quasi continuous. We use Lemma 1.6.7 again It remains to prove that HBe g2 to find g1 with HBe uC 2 Feg1 . If we let g2 D g1 ^ g, then HBe uC n 2 Fe are

4.7 Irreducible recurrence and ergodicity

201

uC 2 L2 .XI g2  m/. Working in the transient extended E g2 -convergent to HBe g 2 uC . Of course, Dirichlet space .F ; E g2 /, we can see the quasi continuity of HBe  the same is true for HBe u . Exercise 4.6.4. Suppose that E is recurrent and that, for u 2 Fe , E.u; u/ D 0 if and only if u D constant. Let B be a nearly Borel set of positive capacity. We put HB D ¹u 2 Fe W E.u; v/ D 0; 8v 2 Fe;X nB º. Show that any u 2 Fe can be expressed uniquely as u D u 0 C u1 ;

u0 2 Fe;X nB ; u1 2 HB ;

u in this case. and e u1 D HBe

4.7

Irreducible recurrence and ergodicity

Throughout this section, we assume that X is a locally compact separable metric space, m is a positive Radon measure on X with full support and M is an m-symmetric Borel right process (a normal strong Markov process with right continuous sample paths and Borel transition function). We denote by .E; F / the Dirichlet form of M on L2 .XI m/ but we do not assume its regularity. The transition function and the resolvent of M are denoted by ¹p t I t  0º and ¹R˛ I ˛ > 0º, respectively, and the L2 -semigroup associated with E is denoted by ¹T t I t > 0º: In §1.5, each T t was extended from L2 \ L1 to a linear operator on L1 .XI m/ satisfying (1.5.2). ¹T t I t > 0º is then a contraction semigroup on L1 .XI m/: We now verify that it is strongly continuous: kT t f  f kL1 .X Im/ ! 0;

t # 0; 8f 2 L1 .XI m/:

On account of (1.5.2), it suffices to show this for any f 2 C0 .X/: Let K be the support of f; G be a relatively compact open set containing K and g be a nonnegative function which equals 1 on X n G and 0 on K: Then Z Z kT t f  f kL1 .X Im/  jp t f .x/  f .x/jm.dx/ C p t jf j.x/m.dx/: G

X nG

The first integral on the Rright-hand side tends to 0 as t # 0; while the second integral is dominated by K p t g.x/jf .x/jm.dx/ which tends to 0 as t # 0: We note that, under the present setting, the concept of an exceptional set introduced in §4.1 still makes sense for M and we use the term ‘q.e.’ to mean ‘except for an exceptional set’. Lemma 4.1.4 remains valid so that, if a finely continuous function for M is non-negative m-a.e., then so it is q.e. A nearly Borel measurable

202

4 Analysis by symmetric Hunt processes

set N  X is said to be m-inessential if m.N / D 0 and X nN is M-invariant in the sense that Px .X t 2 .X n N / [ ¹º; 8t  0/ D 1 for any x 2 X n N: Analogously to Theorem 4.1.1, any exceptional set is contained in some Borel m-inessential set.12 We also note that, if for ˛  0 an ˛-excessive function u is finite m-a.e., then so it is q.e. Indeed, the set N D ¹x 2 X W u.x/ D 1º is then m-negligible and, for x 2 X n N; Ex .e ˛ N u.X N //  u.x/ < 1 so that Px .N < 1/ D 0 and hence N is m-inessential. Theorem 4.7.1. (i) If E is irreducible, then, for any nearly Borel non-exceptional set B, .4:7:1/ Px .B < 1/ > 0; for q.e. x 2 X: (ii) If E is recurrent, then any bounded excessive function u satisfies p t u.x/ D u.x/;

8t > 0; 8x 2 X n N;

.4:7:2/

for some Borel m-inessential set N: (iii) Assume that E is irreducible recurrent. Then, for any nearly Borel nonexceptional set B, Px .B ı n < 1; 8n  0/ D 1;

for q.e. x 2 X:

.4:7:3/

Moreover any excessive function is constant q.e. Proof. (i) If we let Y D ¹x 2 X W Px .B < 1/ > 0º, then m.Y / > 0 by the assumption. On the other hand, (4.6.16) says that X n Y is M-invariant and hence ¹T t º-invariant by Lemma 1.6.1. Therefore m.X n Y / D 0 by the irreducibility assumption. Since X n Y is finely open, it must be exceptional by Lemma 4.1.4. (ii) For a strictly positive function v 2 L1 .XI m/; we have .Rˇ v; u  ˛R˛ u/ D .v; Rˇ u  ˛Rˇ R˛ u/ D .v; R˛ u  ˇR˛ Rˇ u/  .v; R˛ u/ < 1: By letting ˇ ! 0; we get from Lemma 1.6.4 (ii) that ˛R˛ u D u m-a.e. and consequently q.e. Since p t u.x/ is right continuous in t > 0; we get (4.7.2). (iii) We let f .x/ D Px .B < 1/. Then the set E D ¹x 2 X W f .x/ D 1º is T t -invariant. Indeed (4.7.2) implies that f .X t / is a Px -martingale for each x 2 X n N . For any 2 Œ0; 1/, denote by  the hitting time of the finely closed 12 Cf.

Getoor–Sharpe [1].

4.7 Irreducible recurrence and ergodicity

203

set E D ¹x 2 X W f .x/  º. Then, by the optional sampling theorem, we have for any x 2 E n N and T > 0 1 D f .x/ D Ex .f .X  ^T // D Ex .f .X  /I   T / C Ex .f .XT /I  > T /  Px .  T / C Px . > T /; which means that Px .  T / D 0. Hence Px .E c < 1/ D 0;

8x 2 E n N;

.4:7:4/

and E must be T t -invariant in view of Lemma 1.6.1. The set E n N is finely open by (4.7.4) and contains the non-exceptional set B \B r nN: Therefore using Lemma 4.1.4 and the irreducibility again, we conclude that m.X n E/ D 0 and consequently Px .B < 1/ D 1 q.e. by Lemma 4.1.5. (4.7.3) then follows from the Markov property. Finally we take an arbitrary excessive function f . If f were not constant q.e., then we can find 0 < ˛ < ˇ < 1 such that the sets E˛ D ¹x 2 X W f .x/  ˛º and Eˇ D ¹x 2 X W f .x/  ˇº are both non-exceptional. Let g.x/ D f .x/ ^ ˇ, x 2 X. Then g is a bounded excessive function and E˛ D ¹g  ˛º; Eˇ D ¹g D ˇº: By (4.7.3) and the fine continuity of g   Px lim g.X t /  ˛ < ˇ D lim g.X t / D 1 q.e. t !1

t !1

contradicting the fact that g.X t / is a bounded martingale by (4.7.2) and possesses a limit as t ! 1 Px -a.s. for q.e. x 2 X. We note that the last statement of the above theorem holds true for any excessive ˇ function of the restricted Borel right process MˇX nN for any m-inessential set N: Exercise 4.7.1. Show that, under the extra condition of the absolute continuity (4.2.9) for M, the phrase ‘for q.e. x 2 X’ in (4.7.1) and (4.7.3) can be replaced by the stronger one ‘for every x 2 X’. In the rest of this section, we assume that the Dirichlet form E of a given m-symmetric Borel right process M D .X t ; Px / is irreducible recurrent. By Lemma 1.6.5, E is then conservative and the m-symmetry of M implies that m is an invariant measure of ¹p t I t > 0º: Z p t .x; B/m.dx/ D m.B/; 8t > 0; 8B 2 B.X/; .4:7:5/ X

204

4 Analysis by symmetric Hunt processes

0 ; P / induced from which is equivalent to the stationarity of the process .; F1 m the Hunt process M D .; M; X t ; Px / in the following sense:

Pm . t1 ƒ/ D Pm .ƒ/;

0 ƒ 2 F1 :

.4:7:6/

0 is the -field defined by (A.2.1),  is the translation operator (shift) on Here F1 t R 0 : P is a -finite measure on  (§A.2) and Pm D X Px .ƒ/m.dx/; ƒ 2 F1 m 0 .; F1 / but not necessarily a finite measure. Denote by F m the Pm -completion 0 : We say that a statement holds P -a.s. if it holds for all ! 2  n ƒ for of F1 m 0 some ƒ0 2 F m with Pm .ƒ0 / D 0:

Lemma 4.7.1. Suppose u 2 L1C .XI m/ is ¹T t º-invariant, namely, T t u D u; 8t > 0: Then u is constant. Proof. Let u be an m-integrable non-negative Borel function such that p t u .D T t u/ D u; m-a.e. for every t > 0: It suffices to show that u is a constant m-a.e. Using the resolvent of M; define b u.x/ D R1 u.x/; x 2 X: Since ¹T t I t > 0º is a strongly continuous contraction semigroup on L1 .XI m/ it holds then that Z 1 Z 1 e s Ts uds; p t b uD e s T t Cs uds; t > 0; b uD 0

0

where the integrals are taken in Bochner’s sense. By the assumption, we have u Db u m-a.e., t > 0: b u D u m-a.e. p t b u are 1-excessive, they are finely continuous. Further Since both b u and e t p t b u.x/ is right continuous in t  0 for each x 2 X: Therefore there exists an ptb m-inessentialˇ set N such that p t b u.x/ D b u.x/; 8t > 0; 8x 2 X n N: This ˇ means that b u X nN is an excessive function of the Borel right process MX nN : By Theorem 4.7.1 (iii), we conclude that b u D c q.e. for some constant c and u D c m-a.e. Theorem 4.7.2. The stationary process .; F m ; Pm / is ergodic in the following sense: if ˆ 2 L1C .; Pm / is shift invariant, namely, ˆ ı  t D ˆ Pm -a.s.;

t > 0;

.4:7:7/

then ˆ is constant Pm -a.s. 0 -measurable. Define u.x/ D E .ˆ/; x 2 X: Proof. We may assume that ˆ is F1 x For any non-negative bounded v 2 B.X/; we have

0 D Em ..ˆ ı  t  ˆ/v.X0 // D Em ..p t u.X0 /  u.X0 //v.X0 //;

4.7 Irreducible recurrence and ergodicity

205

and we conclude that T t u D u; t > 0; by choosing v suitably. By the preceding lemma, u D c m-a.e. for some constant c and so u.X R t / D c Pm -a.s. for each t > 0: For a positive bounded Borel function h with X hd m D 1; it then holds Phm -a.s. that c D u.Xn / D EXn .ˆ/ D Ehm .ˆ ı n jFn0 / D Ehm .ˆjFn0 / ! ˆ;

n ! 1:

It follows from this that ˆ D c Pm -a.s. We are in a position to formulate an ergodic theorem for M under the present assumption that E is irreducible recurrent. For f 2 L1 .XI m/; we let ´R f d m=m.X/ when m.X/ < 1 .4:7:8/ cf D E 0 when m.X/ D 1: Theorem 4.7.3. Let f be a Borel measurable m-integrable function. (i) It holds Pm -a.s. that Z 1 t f .Xs /ds D cf : lim t !1 t 0

.4:7:9/

(ii) For any non-negative bounded m-integrable function h; the convergence (4.7.9) takes place in L1 .I Phm /: (iii) The convergence (4.7.9) holds Px -a.s. for q.e. x 2 X: (iv) Assume that M satisfies the absolute continuity condition (4.2.9). Let f be a Borel m-integrable function such that for any x 2 X there exists its neighbourhood U.x/ with supy2U.x/ jf .y/j < 1: Then the convergence (4.7.9) holds Px -a.s. for every x 2 X: Proof. (i), (ii) Without loss of generality, we may assume that f is non-negative Borel m-integrable function. Define Z 1 ‰.!/ D f .Xs .!//ds; T ! D 1 !; ! 2 : 0

In view of (4.7.6), T is then a Pm -measure preserving transformation on  and Z 1 hm; ps f ids D kf kL1 .X Im/ ; Em .‰/ D 0

Z ‰.T k !/ D

kC1

f .Xs .!//ds; k

k  1:

206

4 Analysis by symmetric Hunt processes

Z 1 n f .Xs .!//ds; ˆn .!/ D n 0 then, due to the Birkhoff ergodic theorem, If we let

! 2 ;

lim ˆn .!/ D ˆ.!/ Pm -a.s.

n!1

.4:7:10/

for some ˆ 2 L1 .I Pm / and furthermore ¹ˆn º is Pm -uniformly integrable:13 lim sup Em .ˆn I ˆn > r/ D 0:

r!1 n

.4:7:11/

If h satisfies the condition in (ii), then ¹ˆn º is Phm -uniformly integrable, namely, (4.7.11) holds with Ehm in place of Em ; so that the convergence (4.7.10) takes place in L1 .I Phm / because Phm is a finite measure. Obviously, ˆ satisfies the shift invariance (4.7.7) so that ˆ D c Pm -a.s. for some constant c by virtue of Theorem 4.7.2 and furthermore Z 1 t lim f .Xs /ds D c; Pm -a.s: .4:7:12/ t !1 t 0 When m.X/ < 1; (ii) holds for h D 1 and (4.7.12) is also the L1 .I Pm /convergence. Therefore Z 1 n hm; f i D lim hm; ps f ids D cm.X/; n!1 n 0 yielding (4.7.9). When m.X/ D 1; choosing an increasing sequence ¹E` º of Borel sets of finite m-measures increasing to X; we obtain by Fatou’s lemma Z 1 n hm; ps f ids  cPm .X0 2 E` / D cm.E` /; hm; f i D lim inf n!1 n 0 yielding c D 0 D cf by letting ` ! 1: (iii) Assume that f is a non-negative Borel m-integrable function. R1 f is then 1-excessive and finite m-a.e. so that, as was noted above, R1 f .x/ < 1, 8x 2 X n N , for some m-inessential Borel set N: If we let ² ³ Z 1 s e f .Xs /ds < 1 ; 0 D ! 2  W 0

then  t 0  0 and Px .0 / D 1, 8x 2 X n N .ˇ Therefore we may and we do replace  by 0 for the restricted Hunt process MˇX nN : 13 Cf.

Theorem 10.6 and its proof of O. Kallenberg [1].

4.8 Recurrence and Poincaré type inequalities

207

We denote by ƒ0 the set of ! 2 0 for which (4.7.9) is valid. Define g.x/ D Px .ƒ0 /; x 2 X n N Then g D 1 m-a.e. by (i). Clearly ƒ0 D  t1 ƒ0 ; t  0; so that p t g D g; t  0; x 2 X n N; namely, g is an excessive function of MX nN : Therefore g D 1 q.e. (iv) Denote by ƒ the set of ! 2  for which (4.7.9) is valid. By (i), Px .ƒ/ D 1 for every x 2 X n N where N is some Borel set with m.N / D 0: Let n D R 1=n ¹! 2  W 0 f .Xs .!//ds < 1º: For a fixed x 2 X; Px .n / " 1; n ! 1; 1 by the right continuity of X t and the assumption on f: Clearly 1=n ƒ \ n  ƒ: Since Px .X1=n 2 N / D 0 under the absolute continuity condition (4.2.9), we have 1 Px .ƒ/  Px .1=n ƒ \ n / D Ex .PX1=n .ƒ/I n ; X1=n 2 X n N / D Px .n /:

By letting n ! 1; we get Px .ƒ/ D 1: The above theorem remains valid for any m-symmetric general (not necessarily Borel) right process M on a Lusin space whose Dirichlet form is irreducible recurrent by the following reason: if M is a general right process on a Lusin space X which is m-symmetric, then there exists a Borel m-inessential set N such that the restricted process MX nN becomes a special Borel standard process.14 Exercise 4.7.2. Prove the following: (i) If .E; F / is irreducible, then for f 2 L2 .XI m/, lim t !1 T t f D cf in L2 .XI m/ where cf is defined by (4.7.8). (ii) Assume additionally that M satisfies the absolute continuity condition and its transition density is finite, p t .x; y/ < 1; t > 0; x; y 2 X: Then Ex .f .X t // converges to cf for any x 2 X.

4.8

Recurrence and Poincaré type inequalities

Let X be a locally compact separable metric space, m be a positive Radon measure on X with full support and M D .X t ; Px / be an m-symmetric Hunt process on X whose Dirichlet form .E; F / on L2 .XI m/ is regular. ¹p t I t > 0º and ¹R˛ I ˛ > 0º denote the transition function and the resolvent of M, respectively. Throughout this section, E is assumed to be irreducible recurrent. We also assume throughout this section that M satisfies the condition (4.2.15), that is, there exists a Borel properly exceptional set N such that p t .x;  / is absolutely continuous with respect to m for each t > 0 and x 2 X n N . Theorem 4.2.7 14 Cf.

Chen–Fukushima [1; §3.1].

208

4 Analysis by symmetric Hunt processes

provides us with a tractable condition for this: the Sobolev type inequality (4.2.16) holding for the Dirichlet form .E; F / implies the validity of (4.2.15). Indeed the Sobolev type inequality (4.2.16) holds true for a considerably large family of the Dirichlet forms E on account of Theorem 2.4.3. First of all, we make a heuristic observation that the identity Z 1Z 1 Z 1 ˛t 2 ˛R˛ f .x/ D ˛ e p t f .x/dt D ˛ e ˛s p t f .x/dsdt Z D

0

0

1

0

  Z s=˛ s ˛ e p t f .x/dt ds; s 0

t

combined with Theorem 4.7.3 leads us, under the assumption m.X/ < 1, to   1 lim ˛R˛ f  hm; f i .x/ D 0 m-a.e. ˛!0 m.X/ holding for all f 2 L1 .XI m/. In the first part of this section, under the much stronger assumption (4.8.3) stated below, we shall prove the existence of a stronger limit     1 1 R f  hm; f i D lim R˛ f  hm; f i : .4:8:1/ ˛!0 m.X/ m.X/ P n nC1 f for 0 < ˛ < 1 by the resolvent equation, Since R˛ f D 1 nD0 .1  ˛/ R1 (4.8.1) is equal to  R f 

 X   1 1 1 n hm; f i D hm; f i R1 f  m.X/ m.X/

.4:8:2/

nD1

provided that the right-hand side converges. In this case, we shall derive a Poincaré type inequality and show that Rf is a solution of the Poisson equation E.Rf; u/ D R .f; u/ holding for f 2 L2 .XI m/ with X f d m D 0. Generally we can not expect the existence of the limit (4.8.2). In the second half of this section however, we shall employ a simple method of time change and killing studied below in §6.1 and §6.2 to reduce a general case to the above special one in finding a solution Rf of the Poisson equation. We shall then utilize the transient Poincaré type inequality obtained in Theorem 2.4.2 to derive a recurrent Poincaré type inequality. As a consequence, the quotient space FPe of the extended Dirichlet space Fe by its subspace of constant functions will be shown to be a Hilbert space and Rf will be seen to be a representative of a FPe -valued P potential Gf:

4.8 Recurrence and Poincaré type inequalities

209

Exercise 4.8.1. Let .X t ; Px / be the 1-dimensional Brownian motion and r.x; y/ D jxjCjyjjxyj. For any bounded measurable function f with compact support R R such that R1 f .x/dx D 0, show that Rf .x/ D R1 r.x; y/f .y/dy 2 He1 .R1 / and it satisfies 12 D.Rf; u/ D .f; u/ for all u 2 He1 .R1 /; where He1 .R1 / is the space appearing in Exercise 1.6.2. Z

M is said to be Harris recurrent if 1 0

1B .Xs /ds D 1

Px -a.s. for any x 2 X whenever B 2 B.X/; m.B/ > 0:

ˇ Lemma 4.8.1. Under the condition (4.2.15), the restricted Hunt process MˇX nN is Harris recurrent. In particular, for each x 2 X n N and ˛ > 0; we have R˛ .x; B/ > 0 whenever B 2 B.X/; m.B/ > 0: Proof. For any non-negative bounded function g, we get by putting f D g in (4.6.21) R˛g g D R˛ .g.1  R˛g g//  R˛ .g.1  Rg g//: Since R˛g g  1, by letting ˛ to 0, we obtain from Lemma 1.6.4 (iii) that g.1  Rg g/ D 0 m-a.e. In particular, by taking g D 1B for any Borel set B such that m.B/ > 0, we have R1B 1B D R1 a.e. on B. Using the relation R1B 1B .x/ D t Ex .1  e C1 / holding for C t D 0 1B .Xs /ds, this implies that Px .C1 D 1/ D 1 for m-a.e. x 2 B. Since Ex .PX t .C1 D 1// D Px .C1 D 1/ for any t > 0, Px .C1 D 1/ is a T t -invariant function. Hence, by virtue of Lemma 4.7.1, Px .C1 D 1/ D 1 m-a.e. x 2 X and consequently for all x 2 X n N from the assumption (4.2.15). This implies the Harris recurrence of MjX nN . Since p t .x;  / is absolutely continuous with respect to m for all x 2 X n N , Lemma 4.2.4 yields that there exists a non-negative jointly measurable function r˛ .x; y/ satisfying R˛ .x; dy/ D r˛ .x; y/m.dy/, r˛ .x; y/ D r˛ .y; x/ and Z r˛ .x; z/rˇ .z; y/m.dz/ D 0 r˛ .x; y/  rˇ .x; y/ C .˛  ˇ/ X

for all x; y 2 X n N . Moreover, r˛ .  ; x/ and hence r˛ .x;  / are ˛-excessive functions for each fixed x 2 X n N . First we assume that the measure m and the density r1 .x; y/ satisfy the following additional condition: m.X/ < 1 and r1 .x; y/ > c for some c > 0 and all x; y 2 X n N : .4:8:3/ R Note that cm.X/ < X r1 .x; y/m.dy/  1. For a signed measure , denote by kk its total variation. Further put D 1  cm.X/=2 < 1 and m1 .dx/ D

210

4 Analysis by symmetric Hunt processes

m.dx/=m.X/. The Lp -norm of f 2 Lp .XI m/ will be denoted by kf kp for 1  p  1. Lemma 4.8.2. Assume that (4.8.3) holds. Then, for any n  1, sup x;x 0 2X nN

k.R1 /n .x;  /  .R1 /n .x 0 ;  /k  2 n :

.4:8:4/

Furthermore, if f 2 L2 .XI m/ then p k.R1 /n f  hm1 ; f ik2  2. /n kf  hm1 ; f ik2 :

.4:8:5/

Proof. For any x; x 0 2 X n N , put F  Fx;x 0 D ¹y W r1 .x; y/  r1 .x 0 ; y/ > 0º. Then kR1 .x;  /  R1 .x 0 ;  /k Z Z 0 .r1 .x; y/  r1 .x ; y//m.dy/ C D

X nF

F

.r1 .x 0 ; y/  r1 .x; y//m.dy/

 2  cm.X/ D 2 : This implies (4.8.4) for n D 1. To show (4.8.4) for general n  1, for any kernels P1 ; P2 such that Pi .x; X/ D Pi .x; X n N / D 1 for x 2 X n N and i D 1; 2, put Q.Pi / D

1 sup kPi .x;  /  Pi .x 0 ;  /k: 2 x;x 0 2X nN

Then it suffices to show the inequality Q.P1 P2 /  Q.P1 /Q.P2 / . Let X C and X  be the sets which support the positive and negative parts of P1 P2 .x;  /  P1 P2 .x 0 ;  /, respectively. Then kP1 P2 .x;  /  P1 P2 .x 0 ;  /k D P1 P2 .x; X C /  P1 P2 .x 0 ; X C /  P1 P2 .x; X  / C P1 P2 .x 0 ; X  / Z P1 .x; dy/.P2 .y; X C /  P2 .y; X  // D X nN

Z

 Z D

X nN

X nN

P1 .x 0 ; dy/.P2 .y; X C /  P2 .y; X  //

.P1 .x; dy/  P1 .x 0 ; dy//.P2 .y; X C /  P2 .y; X  /  a/

4.8 Recurrence and Poincaré type inequalities

211

for any constant a. In particular, for p.y/ D P2 .y; X C /  P2 .y; X  /, let a D 1 2 .supy2X nN p.y/ C infy2X nN p.y//. Then we have the estimate sup jp.y/  aj 

y2X nN

1 sup jp.y/  p.y 0 /j 2 y;y 0 2X nN

1 sup jP2 .y; X C /  P2 .y 0 ; X C / C P2 .y 0 ; X  /  P2 .y; X  /j 2 y;y 0 2X nN



 Q.P2 /: Therefore sup x;x 0 2X nN

kP1 P2 .x;  /  P1 P2 .x 0 ;  /k  Q.P2 /

sup x;x 0 2X nN

kP1 .x;  /  P1 .x 0 ;  /k

D 2Q.P1 /Q.P2 /: .n/

For the proof of (4.8.5), let r1 .x; y/ be the density of .R1 /n .x; dy/ relative to m.dy/. By using the notation f0 .x/ D f .x/  hm1 ; f i, we have from the .n/ .n/ symmetry r1 .x; y/ D r1 .y; x/ and the identity .R1 /n 1.x/ D 1; x 2 X n N; that k.R1 /n f  hm1 ; f ik22 D k.R1 /n f0 k22 ³2 Z ²Z Z .n/ .n/ 0 0 D .r1 .x; y/  r1 .x ; y//f0 .y/m.dy/m1 .dx / m.dx/ X

X

X

X

X

X

Z ²Z Z 

 

jr1 .x; y/  r1 .x 0 ; y/jm.dy/m1 .dx 0 / .n/

Z Z X

sup x;x 0 2X nN

X

 4

.n/ jr1 .x; y/

³



.n/ r1 .x 0 ; y/jf0 .y/2 m.dy/m1 .dx 0 /

m.dx/

k.R1 /n .x;  /  .R1 /n .x 0 ;  /k Z Z Z 

n

.n/

X

X

X

.r1 .x; y/ C r1 .x 0 ; y//f02 .y/m.dy/m1 .dx 0 /m.dx/ .n/

.n/

kf0 k22 ;

yielding (4.8.5). By virtue of Lemma 4.8.2, we can define a linear operator R from L2 .XI m/ to it by Rf D

1 X nD1

..R1 /n f  hm1 ; f i/ D lim .R˛ f  hm1 ; f i/; ˛!0

f 2 L2 .XI m/;

212

4 Analysis by symmetric Hunt processes

and we get from (4.8.5) the bound p 2 kRf k2  p kf  hm; f ik2 ; 1

f 2 L2 .XI m/:

R .n/ .n/ Since .R1 /n f .x/  hm1 ; f i D X .R1 .x; dy/  R1 .x 0 ; dy//f .y/m1 .dx 0 /, (4.8.4) implies, for any f 2 L1 .XI m/, k.R1 /n f  hm1 ; f ik1  2 n kf k1 ;

kRf k1 

2 kf k1 ; 1

.4:8:6/

namely, R is also a linear operator from L1 .XI m/ to it. In order to obtain a little finer L2 estimate of R, we consider any f 2 L2 .XI m/ with hm; f i D 0 and we let, for an integer k and t > 0, .ˆ2k f /.t; x/ D .R1 /k f .x/ and .ˆ2kC1 f /.t; x/ D .R1 /k p t =2 f .x/. Then ˆn Rf D Rˆn f and we get from the above Z 1 n e t .ˆn f .t;  /; Rˆn f .t;  // dt ..R1 / f; Rf / D 0

p Z 1 2 e t kˆn f .t;  /k22 dt;  p 1 0

n  1:

Since kˆ2k f .t;  /k2 D .f; .R1 /2k f / and kˆ2kC1 f .t;  /k2 D .f; .R1 /2k p t f /, we have p 2 n ..R1 / f; Rf /  p .f; .R1 /n f /; n  1: 1 By taking sum in n  1, we obtain the inequality kRf k22  any f 2 get the bound

L2 .XI m/;

kRf

k22

p 2  p .f; Rf 1 

/: For

we substitute f  hm1 ; f i in this inequality in place of f to

 p 2 p 2 2  kf hm1 ; f ik22 : .4:8:7/ p .f  hm1 ; f i; Rf /  p 1 1

Along with the estimates (4.8.6) and (4.8.7), the following is a main result under the condition (4.8.3). (4.8.9) will be called a recurrent Poincaré type inequality. Theorem 4.8.1. Assume that condition (4.8.3) is satisfied. Then, for any f 2 L2 .XI m/ such that hm; f i D 0, Rf 2 F and E.Rf; u/ D .f; u/

.4:8:8/

213

4.8 Recurrence and Poincaré type inequalities

for all u 2 F . Furthermore, for any u 2 F , ku 

hm1 ; uik22

p 2  p E.u; u/: 1

.4:8:9/

Proof. For any f 2 L2 .XI m/ such that hm; f i D 0, R.n/ f D

n X

.R1 /k f 2 F

kD1

and Rf D limn!1 R.n/ f m-a.e by taking a subsequence if necessary. Since E.R.n/ f; R.n/ f / D E1 .R.n/ f; R.n/ f /  .R.n/ f; R.n/ f / D

n X

.f; .R1 /kCm1 f  .R1 /kCm f /

k;mD1

D

n X

.f; .R1 /k f  .R1 /kCn f /;

kD1

is uniformly bounded relative to n, Rf belongs to F D Fe \ L2 .XI m/ and E.Rf; Rf / D .f; Rf /. The relation (4.8.8) follows similarly. For any u 2 F , we have from (4.8.8) ku  hm1 ; uik22 D E.R.u  hm1 ; ui/; u  hm1 ; ui/  E.R.u  hm1 ; ui/; R.u  hm1 ; ui//1=2  E.u  hm1 ; ui; u  hm1 ; ui/1=2 D .u  hm1 ; ui; R.u  hm1 ; ui//1=2  E.u  hm1 ; ui; u  hm1 ; ui/1=2 p 2 1=2 . p / ku  hm1 ; uik2 E.u; u/1=2 1 which gives (4.8.9). Now we return to the general case that, for a certain properly exceptional set N  X; p t .x;  / is absolutely continuous with respect to m for each x 2 X n N: Let r˛ .x; y/ be the density function of the resolvent R˛ .x; dy/ with respect to m.dy/ specified in the paragraph after the proof of Lemma 4.8.1. Lemma 4.8.3. (i) There exist compact sets F; K  X n N and a positive constant a > 0 such that 0 < m.F / < 1; 0 < m.K/ < 1 and r1 .x; y/ > a for all x 2 F and y 2 K.

214

4 Analysis by symmetric Hunt processes

(ii) There exists a positive constant c and a nearly Borel measurable finely closed set B  X n N such that 0 < m.B/ < 1 and r1 .x; y/ > c for all x; y 2 B. Proof. (i) Let a D ¹.x; y/ 2 .X n N /  .X n N / W r2 .x; y/ > aº for a  0: By Lemma 4.8.1 and the symmetry of r2 ;Rwe have, for any x; y 2 X nN , r2 .x; z/ > 0, r2 .z; y/ > 0 m-a.e. and r1 .x; y/  r2 .x; z/r2 .z; y/m.dz/ > 0: Consequently 0 D .X n N /  .X n N /: Take a compact set A  X n N with m.A/ > 0 and set ƒa D .A  A/ \ a : We can then find a > 0 such that m  m.ƒa / > 0: For ƒ  .X n N /  .X n N / and x 2 X n N; we denote by ƒ.x/ the x-section of ƒ: S .n/ .nC1/ º is a reLet A D i Ai be a sequence of partitions of A such that ¹Ai .n/ finement of ¹Ai º and ƒa belongs to the -field generated by direct products of elements in the sequence. For x; z 2 A; denote by i.x/; i.z/ the indices of the cells containing them, respectively, we see from the differentiation theorem that, there exists a set   A  A with m  m./ D 0 and .n/

.n/

.m  m/.ƒa \ .Ai.x/  Ai.y/ // .n/

.n/

.m  m/.Ai.x/  Ai.y/ /

! Iƒa .x; y/;

n!1

for .x; y/ 2 .A  A/ n : If we let B D ¹z 2 A W m.ƒa .z/ n .z// > 0º; then m.B/ > 0: Therefore, for z0 2 B and x0 ; y0 such that x0 ; y0 2 ƒa .z0 / n .z0 /; we can choose n with .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.m  m/.ƒa \ .Ai.x0 /  Ai.z0 / // > .3=4/.m  m/.Ai.x0 /  Ai.z0 / /; .m  m/.ƒa \ .Ai.y0 /  Ai.z0 / // > .3=4/.m  m/.Ai.y0 /  Ai.z0 / /: The first inequality in the above and the Fubini theorem yields m.F / > 0 for the set .n/ .n/ .n/ F D ¹x 2 Ai.x0 / W m.Ai.z0 / \ ƒa .x// > .3=4/m.Ai.z0 / /º: Similarly the second inequality yields m.K/ > 0 for the set .n/

.n/

.n/

K D ¹y 2 Ai.y0 / W m.Ai.z0 / \ ƒa .y// > .3=4/m.Ai.z0 / /º: It holds for x 2 F and y 2 K that .n/

m.ƒa .x/ \ ƒa .y//  m.ƒa .x/ \ ƒa .y/ \ Ai.z0 / / .n/

.n/

.n/

 m.Ai.z0 / \ ƒa .x// C m.Ai.z0 / \ ƒa .y//  m.Ai.z0 / / 1 .n/ > m.Ai.z0 / / > 0 2

4.8 Recurrence and Poincaré type inequalities

215

and consequently Z r1 .x; y/ 

ƒa .x/\ƒa .y/

r2 .x; z/r2 .z; y/m.dz/ 

a2 .n/ m.Ai.z0 / /: 2

F and K can be taken to be compact due to the inner regularity of the measure m: (ii) By the irreducibility of M and Theorem 4.7.1 (i), we have for the set F; K of (i) Ex .e  F / > 0 and Ex .e  K / > 0 for q.e. x 2 X. Hence there exists a constant b > 0 such that the set B D ¹x 2 X nN W Ex .e  F /  b; Ex .e  K /  bº is of positive m-measure. As the functions involved are 1-excessive, B is nearly Borel and finely closed. Since r1 .  ; x/ and r1 .x;  / are 1-excessive, we have for any x; y 2 B r1 .x; y/  Ex .e  F r1 .X F ; y//  Ex .e  F Ey .e  K r1 .z; X K //jzDXF / > a Ex .e  F /Ey .e  K /  ab 2 : It suffices to put c D ab 2 . By replacing B by its intersection with a certain relatively compact open set, m.B/ can be also assumed to be finite. Let us fix a nearly Borel finely closed subset B of X n N such that 0 < m.B/ < 1 and 0 < c < r1 .x; y/; x; y 2 B; according to Lemma 4.8.3 (ii). To proceed further, we apply the results in §6.1 and §6.2 on the killing and time change of M by means of the simple positive continuous additive functional Rt C t D 0 1B .Xs /ds; t  0; with Revuz measure mB : L D .X t ; Px / be the time changed process of MjX nN where  t is the right Let M L is a right process with state space being the support continuous inverse of C t : M e e e is a quasi support of mB and B is quasi closed. B of C t : B  B q.e. because B  e Denote by B . B/ the topological support of B: According to Theorem 6.2.1 and L FL / L is mB -symmetric and recurrent, and its Dirichlet form .E; Theorem 6.2.3, M 2 2  on L .BI mB / D L .B I mB / is regular admitting the expression (6.2.4) with e in place of ; Y; Y e: In particular, mB ; B  ; B L E.'; '/ D E.H '; H '/;

' 2 FL D Fe jB \ L2 .BI mB /;

.4:8:10/

L where H '.x/ D Ex .'.X /I  e B < 1/; x 2 X: Let ¹pL t I t  0º; ¹Rp I p > 0º e B L respectively. As M L is recurrent, be the transition function and the resolvent of M, it is conservative and RL 1 1B D 1B : In view of (6.2.5), we have the identity H.RLp '/.x/ D RpC .1B  '/.x/;

q.e. x 2 X; ' 2 BC .B/:

.4:8:11/

216

4 Analysis by symmetric Hunt processes

Here 1B ' denotes the zero extension of ' from B to X and  Z 1 pC pC ˛t pC t e f .X t /dt ; x 2 X; RpC D R0 ; R˛ f .x/ D Ex 0

the resolvent of the subprocess MpC of M by the multiplicative functional e pC t , which is m-symmetric with the Dirichlet form on L2 .XI m/ given by E pmB .u; v/ D E.u; v/ C p.u; v/mB ;

u; v 2 F pmB D F \ L2 .mB / .D F /

in view of Theorem 6.1.1. Due to the irreducibility assumption on M; we can see that this Dirichlet form is transient in view of the proof of Lemma 6.2.6 (i). If a Borel function f satisfies .jf j; RC jf j/ < 1; then by Theorem 1.5.4, RC f 2 FemB ;

8v 2 FemB ; .4:8:12/ mB m m B B where Fe denotes the extended Dirichlet space of .E ; F / which equals Fe \ L2 .BI mB / by virtue of Lemma 6.2.5. Since R1C .x;  /  R1 .x;  /, R1C .x;  / and hence RC .x;  / is absolutely continuous relative to m for x 2 X n N . Therefore, RL 1 .x;  / is absolutely continuous e \ .X n N /. As in Lemma 4.2.4, we can take a density relative to mB for any x 2 B e function rL1 .x; y/; x; y 2 B; to be symmetric and 1-excessive in each variable relL Since R1 .x; A \ B/ e  RL 1 .x; A \ B/ e for any x 2 B e and any Borel set ative to M: e A, it follows that c  rL1 .x; y/ for any x; y 2 B \ .X n N /. Hence the condition L (4.8.3) is fulfilled by M: Therefore by putting D 1  cm.B/=2, we can conclude from Lemma 4.8.2, (4.8.6) and (4.8.7) that, if we define the operator RL by L D R'

f  v 2 L1 .XI m/;

1  X .RL 1 /n '  nD1

E mB .RC f; v/ D .f; v/;

 1 hmB ; 'i ; m.B/

' 2 L2 .BI mB /;

then RL is a linear operator making the spaces L1 .BI mB / and L2 .BI mB / invariant and satisfying the bounds L 1  2 k'k1 ; ' 2 L1 .BI mB /; kR'k 1  p  2 1 2 L L kR'kmB  ' hmB ; 'i; R' p 1 m.B/ mB  p 2 2 1 hmB ; 'ik2mB ;  k'  p 1 m.B/

(4.8.13)

' 2 L2 .BI mB /: (4.8.14)

4.8 Recurrence and Poincaré type inequalities

217

Moreover, by Theorem 4.8.1, we have for any ' 2 L2 .BI mB / with hmB ; 'i D 0; L 2 FL ; R'

L R'; L E. / D .'; /mB ;

2 FL :

.4:8:15/

We next prove that mB RC D m:

.4:8:16/

In fact we have from (4.4.9) and the symmetry of Rˇ and RC 0 D .g; Rˇ 1B  RC 1B C ˇRˇ RC 1B  Rˇ .1B RC 1B // D .Rˇ g  RC g C ˇRC Rˇ g  Rˇ g; 1B / D hmB RC ; g  ˇRˇ gi: Denote by h the density of mB RC with respect to m: h is then ¹T t º-invariant by the above and a constant by Lemma 4.7.1. Since mB RC .B/ D m.B/, h D 1: Define now a recurrent potential Rf of a function f on X by L B RC f C RC 1B RC f C RC f  Rf D RC 1B R1

2 hm; f i: m.B/

D.R/ D ¹f 2 L1 .XI m/ \ L1 .XI m/ W .jf j; RˇC jf j/ < 1; RC jf jˇB 2 L1 .BI mB /º:

(4.8.17)

(4.8.18)

Take f 2 D.R/: Then Rf 2 FemB because of (4.8.12), (4.8.13) and (4.8.17). L D 0 for ' 2 Further hmB ; Rf i D 0 on account of (4.8.16) and hmB ; R'i 2 L .BI mB /: Theorem 4.8.2. (i) If f 2 D.R/ and hm; f i D 0; then Rf 2 FemB and E.Rf; u/ D .f; u/

.4:8:19/

for all u 2 Fe . (ii) There exists a strictly positive bounded m-integrable function g with kRC gk1 < 1; g  1B such that a recurrent Poincaré type inequality ˇ2 p Z ˇ ˇ ˇ ˇu.x/  1 hmB ; uiˇ g.x/m.dx/  4.3 p / kRC gk1 E.u; u/ ˇ ˇ m.B/ 1 X .4:8:20/ is valid for all u 2 Fe . Proof. We first note that it is sufficient to prove (i) and (ii) holding for all u 2 FemB : To see this, assume that (4.8.20) is true for any u 2 FemB : Then it holds for any u 2 F : Take any u 2 Fe and its approximating sequence ¹un º  F : Then

218

4 Analysis by symmetric Hunt processes

(4.8.20) holding for un implies that un 

1 hmB ; un i m.B/

is L2 .g  m/-convergent

1 hmB ; un i ! k, to some v 2 L2 .g  m/ as n ! 1: Since un ! u m-a.e., m.B/ n ! 1; m-a.e. for some constant k by taking a subsequence if necessary, so that u D v C k 2 L2 .g  m/ and u 2 L2 .mB / because g  1B : This means that actually Fe D FemB : (i) For any f 2 D.R/ with hm; f i D 0; we have ' D RC f jB 2 L1 .BI mB / and hmB ; 'i D 0: It follows from (4.8.11) and (4.8.17) that

L C RL 1 ' D R'; L .Rf  RC f /jB D RL 1 R'

L Rf  RC f D H R':

For any u 2 FemB ; ujB 2 FL in view of (4.8.10). Therefore (4.8.10), (4.8.11), (4.8.12) and (4.8.15) yield that L E.Rf; u/ D E..Rf  RC f /jB ; ujB / C E.RC f; u/ D .'; u/mB C E.RC f; u/ D E mB .RC f; u/ D .f; u/: (ii) Take a strictly positive function g 2 L1 .XI m/ \ L1 .XI m/ such that kRC gk1 < 1. To show the existence of such function, let h be a bounded strictly positive function of L1 .XI m/ such that RC h < 1. SFor an increasing sequence of closed sets ¹Cn º such that m.Cn / < 1 and n Cn D X, let An D ¹x 2 Cn W RC h.x/  nº. Using the hitting probability HACn of An relative to the subprocess MC , we have RC .1An h/ D HACn RC .1An h/  n. Hence P 1 n C it is enough to put g D 1 nD1 n.m.An /_1/ 2 1An h. Since R 1B  1; we may assume that g  1B by replacing g with g C 1B if necessary. Let f be a Borel function such that jf j  kg for some k > 0 and hm; f i D 0. Then f 2 D.R/ and Rf 2 FemB : Since E mB is a transient Dirichlet form, we get by Theorem 2.4.2 and (4.8.19) Z .Rf /2 g d m  4kRC gk1 E mB .Rf; Rf / X

2 D 4kRC gk1 Œ.f; Rf / C kRf kL 2 .m / : B

L C ' for ' D RC f jB ; we obtain from (4.8.14) Since Rf jB D R' 2 2 L 2 L kRf kL 2 .m / D kR'kL2 .m / C 2.R'; '/mB C k'kL2 .m / B B B



2 2 L '/mB C k'k2 2  p .R'; p .Rf; RC f /mB : L .m / B 1 1

As E.Rf; RC f / D .f; RC f / D E mB .RC f; RC f /  0; the last expression in the above is dominated by 12p E mB .Rf; RC f / D 12p .f; Rf /: Therefore we

4.8 Recurrence and Poincaré type inequalities

have

219

p 4.3  / kRC gk1 E.Rf; Rf /; .Rf / .x/g.x/m.dx/  p 1 X

Z

2

proving (4.8.20) for u D Rf as hmB ; Rf i D 0: mB We next let Fe;0 D ¹u 2 FemB W hmB ; ui D 0º and prove (4.8.20) for mB any u 2 Fe;0 . It suffices to show that the family R D ¹Rf W jf j  kg mB mB : Since Fe;0 is an E mB for some k > 0 , hm; f i D 0º is E mB -dense in Fe;0 mB must closed subspace of FemB , it is enough to show that a function u 2 Fe;0 m vanish under the assumption E B .Rf; u/ D 0 for all Rf 2 R: b D f  1 hm; f i1B for any measurable Under this assumption, we put f m.B/ b 2 R so that, by (4.8.19), function f satisfying jf j  kg. Then Rf b; u/ D .f b; u/ C .Rf b; u/m D .f b; u/ C .R' L C '; 1B u/mB ; 0 D E mB .Rf B ˇ bˇ : Since hmB ; 'i D 0; u 2 L2 .BI mB / and hmB ; ui D 0; it where ' D RC f B holds that L D R'

1 X

.RL 1 /n ';

L B u/ D R.1

nD1

1 X

.RL 1 /n .1B u/:

.4:8:21/

nD1

By the symmetry of RL 1 on L2 .BI mB / and the symmetry of RC on L2 .XI m/; we then get b; u/ C .RC f b; 1B R.1 L B u/ C 1B u/ D .f; u C RC .1B R.1 L B u/ C 1B u//: 0 D .f L B u/ C 1B u/ D 0: But then (4.8.11) and (4.8.21) imply Therefore u C RC .1B R.1 L B u/ C 1B u D 1B .u C RC .1B R.1 L B u/ C 1B u// D 0; 1B R.1 and we arrive at the desired conclusion that u D 0: mB We have shown that (4.8.20) holds for any u 2 Fe;0 : But it then holds for any u 2 FemB as we may substitute v D u  Let K.g/ D . the following

hmB ;ui 1 .2 m.B/ B

p 4.3  / p kgkL1 .X Im/ kRC gk1 /1=2 . 1 

L1 -estimate

mB Fe;0 / in (4.8.20).

Then, by virtue of (4.8.20),

holds.

Corollary 4.8.1. It holds for all u 2 Fe that ˇ Z ˇ ˇ ˇ ˇu.x/  1 hmB ; uiˇg.x/m.dx/  K.g/E.u; u/1=2 : ˇ ˇ m.B/ X

.4:8:22/

220

4 Analysis by symmetric Hunt processes

(4.8.20) implies for u 2 Fe that E.u; u/ D 0 if and only if u is constant. We can further deduce the following from (4.8.20) in the same way as Lemma 1.5.5 for the transient case. Corollary 4.8.2. The quotient space FPe of Fe by the subspace of constant functions is a Hilbert space with inner product E: Any E-Cauchy sequence ¹un º  Fe admits u 2 Fe and constants cn such that un is E-convergent to u and un C cn is L2 .XI g  m/-convergent to u as n ! 1: Consider the space L0 D ¹f 2 L1 .XI m/ W hm; f i D 0; .jf j; RC jf j/ < 1º: For each f 2 L0 ; ˆf .u/ D .u; f /;

u 2 Fe .D FemB /;

defines a linear functional on the quotient space FPe on account of (4.8.12). Let L D ¹f 2 L0 W ˆf is boundedº: A function f 2 L0 belongs to L if and P 2 FPe uniquely such that only if there exists Gf P u/ D .f; u/; E.Gf;

u 2 FPe :

Theorem 4.8.2 (i) states that, if f 2 L0 is bounded on X and RC jf j is bounded P with on B; then f 2 L and Rf defined by (4.8.17) is a representative of Gf P hmB ; Rf i D 0: The proof of Theorem 4.8.2 (ii) implies that G.L/ is an E-dense P P subspace of Fe : We may call Gf a recurrent potential of f 2 L:

Chapter 5

Stochastic analysis by additive functionals

We assume throughout this chapter that we are given a symmetric Hunt process whose Dirichlet form is regular. It is shown in §5.1 that all equivalence classes of PCAF’s of the Hunt process are in one-to-one correspondence with all smooth measures introduced in §2.2. Such a correspondence is useful in that it enables us to reduce the calculus of functionals to the calculus of measures. After §5.2, we are concerned with additive functionals which are not necessarŒu ily non-negative. In particular, the AF A t D e u.X t /  e u.X0 / generated by the function u in the Dirichlet space admits a unique decomposition AŒu D M Œu C N Œu ; where M Œu is a martingale AF of finite energy and N Œu is a continuous additive Œu functional of zero energy (§5.2). N t is not of bounded variation in t unless u is a potential of a signed smooth measure in a certain sense (§5.4 and §5.5). Thus the decomposition in §5.2 may be regarded as a generalization (towards the Dirichlet space) of the Doob–Meyer decomposition of supermartingales and Itô’s formula on semi-martingales. The Beurling–Deny formula of §3.2 results from a further decomposition of M Œu (§5.3). Several methods of computing and studying M Œu ; N Œu and their extended versions are provided in §5.2, §5.4 and §5.5. The LeJan formula of §3.2 is translated into a transformation rule involving stochastic integrals (§5.6). In §5.7, the Lyons–Zheng decomposition of AŒu into a sum of forward and backward martingale AF’s will be presented, which will then be applied to the study of the conservativeness and tightness for symmetric diffusions. AF’s in the strict sense are also investigated under the absolute continuity condition for the transition function at the ends of §5.1, §5.2 and §5.5. Most sections contain examples concerning the Brownian motion and the diffusions associated with second order differential operators of divergence form.

222

5.1

5 Stochastic analysis by additive functionals

Positive continuous additive functionals and smooth measures

Throughout Chapter 5, let .X; m/ be as in (1.1.7) and M D .; M; X t ; Px / be an m-symmetric Hunt process on X whose Dirichlet form .E; F / on L2 .XI m/ is regular.1 Potential theories relevant to .E; F / and M have been developed in Chapter 2 and Chapter 4. We now utilize these theories and study the structures of some important classes of additive functionals of the Hunt process M.2 From now on in the main text, we shall call an additive functional of M in the sense of Appendix A.2 and A.3 an additive functional in the strict sense in order to distinguish it from the somewhat relaxed notion we are going to introduce. Let us call an extended real valued function A t .!/; t  0; ! 2 , an additive functional (AF in abbreviation) if A t .!/ is an additive functional in the strict sense but with respect to the restricted Hunt process MjX nN ; N being a properly exceptional set which depends on A in general. More precisely, in order for A t .!/ to be an AF the following conditions must hold: (A.1) A t .  / is F t -measurable, ¹F t º being the minimum completed admissible filtration (see §A.2); (A.2) there exist a set ƒ 2 F1 and an exceptional set N  X such that Px .ƒ/ D 1, 8x 2 X n N ,  t ƒ  ƒ, 8t > 0, and moreover, for each ! 2 ƒ, A:.!/ is right continuous and has the left limit on Œ0; .!//, A0 .!/ D 0, jA t .!/j < 1, 8t < .!/, A t .!/ D A .!/ .!/, 8t  .!/ and A t Cs .!/ D As .!/ C A t .s !/, 8t; s  0. The sets ƒ and N referred to in the preceding paragraph are called a defining set and an exceptional set of the AF A t .!/ respectively. .1/ Two AF’s A.1/ and A.2/ are said to be equivalent if for each t > 0 Px .A t D .2/ A t / D 1 q.e. x 2 X. By virtue of Theorem 4.1.1, we can then find a common defining set ƒ and a common properly exceptional set N of A.1/ and A.2/ such .1/ .2/ that A t .!/ D A t .!/; 8t > 0; 8! 2 ƒ. An AF A t .!/ is said to be finite (resp. continuous) if jA t .!/j < 1; 8t 2 Œ0; 1/ (resp. A t .!/ is continuous in t 2 Œ0; 1// for any ! in a defining set. We also call A t .!/ cadlag if A t .!/ is right continuous and possesses the left limit on Œ0; 1/ for any ! in a defining set. A Œ0; 1-valued continuous AF is called a positive continuous AF (PCAF in abbreviation). The set of all PCAF’s is 1 As

for a reduction of the case where X is a Lusin space to the present setting, see §7.3 and Theorem A.4.3 of the Appendix A.4. 2 The convention made in the last chapter is still in force in this chapter: any numerical function f on X is extended to X by setting f ./ D 0.

5.1

223

Positive continuous additive functionals and smooth measures

C denoted by AC c : The aim of this section is to characterize Ac by the class S of smooth measures associated with .E; F / (see §2.2). To this end we first construct a PCAF for any  2 S0 – the positive Radon measure of finite energy integral. After establishing the relation of AC c and S , it is not difficult to give an analytical characterization of the family of all PCAF’s in the strict sense, as we shall do at the end of this section under the absolute continuity assumption of the transition function.

Lemma 5.1.1. (i) For any u 2 F ;  2 S0 ; 0 < T < 1 and " > 0, P



p  eT p u.X t /j > "  E1 ./ E1 .u; u/: sup je " 0t T

.5:1:1/

.0/

(ii) Suppose that E is transient. For any u 2 Fe ;  2 S0 and " > 0 P



p  1p sup je E./ E.u; u/: u.X t /j > "  " 0t "º. Then the left-hand side of (5.1.1) is dominated by e T X p.x/.dx/ with p.x/ D Ex .e  E /. By virtue of Theorem 2.1.5 and Theorem 4.2.5, we have Z p p p.x/.dx/ D E1 .p; U1 /  E1 ./ Cap.E/ X



p 1p E1 ./ E1 .u; u/: "

(ii) can be proved analogously to the above by making use of the 0-order version of Theorem 2.1.5 together with Theorem 4.3.3. Lemma 5.1.2. (i) Let ¹un º be a sequence of quasi continuous functions in F . If ¹un º is an E1 -Cauchy sequence, then there exists a subsequence ¹unk º satisfying the condition that for q.e. x 2 X Px .unk .X t / converges uniformly in t on each compact interval of Œ0; 1// D 1: (ii) Suppose that E is transient. Let ¹un º be an E-Cauchy sequence of quasi continuous functions in Fe . There exists then a subsequence ¹unk º satisfying Px .unk .X t / converges uniformly in t 2 Œ0; 1// D 1

q.e. x 2 X:

224

5 Stochastic analysis by additive functionals

p Proof. (i) Take ¹nk º such that E1 .unkC1  unk ; unkC1  unk /  22k . By p virtue of Lemma 5.1.1 (i), P .ƒk /  e T 2k E1 ./ with ƒk D ¹sup0t T junk .X t /  unkC1 .X t /j > 2k º. By the Borel–Cantelli lemma, the P -measure of the set ƒ D limk ƒk vanishes for every  2 S0 . Hence, Px .ƒ/ D 0 q.e. by virtue of Theorem 2.2.3. (ii) can be proved similarly to the above by using Lemma 5.1.1 (ii) and the 0-order version of Theorem 2.2.3. Henceforth we use the notations h; ui D hu; i D  and a function u.

R X

u.x/.dx/ for a measure

C Theorem R 1 t5.1.1. For any  2 S0 , there exists a finite A 2 Ac such that Ex . 0 e dA t / is a quasi continuous version of U1 .

Proof. Take a non-negative finite Borel and quasi continuous version u of U1  such that ´ nRnC1 u.x/ " u.x/; n ! 1; 8x 2 X n N; .5:1:3/ u.x/ D 0; 8x 2 N; where N is some properly exceptional set and ¹R˛ ; ˛ > 0º is the resolvent of the process M. We let ´ n .u.x/  nRnC1 u.x// ; x 2 X n N gn .x/ D 0; x 2 N: Then it is easy to see that gn  m !  vaguely, R1 gn .x/ " u.x/; 8x 2 X n N , and, moreover, R1 gn is E1 -convergent to u. We define the approximating 1-order PCAF e An of M by e An .t; !/ D

Z

t

e s gn .Xs .!// ds:

0

Then, for any  2 S00 E ..e An .1/  e Al .1//2 /  2M

p

E1 ./kR1 gn  R1 gl kE1 3

.5:1:4/

with M D kU2 k1 . In fact, by setting gn;l D gn  gl ; n > l, the left-hand side 3 kvk E1

D

p

E1 .v; v/

5.1

Positive continuous additive functionals and smooth measures

of (5.1.4) is equal to Z 1 Z e s gn;l .Xs /ds 2E 0

D 2E

1

e u gn;l .Xu /du

s

Z

1

e 2s gn;l .Xs /R1 gn;l .Xs /ds

225





0

D 2h; R2 .gn;l R1 gn;l /i D 2.U2 ; gn;l R1 gn;l /  2.U2 ; gn R1 gn;l /  2M .gn ; R1 gn  R1 gl / D 2M E1 .R1 gn ; R1 gn  R1 gl /: Applying the Schwarz inequality and noting E1 .R1 gn ; R1 gn / D .gn ; R1 gn /  .gn ; u/ D h; R1 gn i  h; ui D E1 ./; we obtain (5.1.4). An .1/jF t / D e An .t/ C e t EX t .e An .1// D e An .t / C e t R1 gn .X t /, Since E .e we see that Mn .t/ D e An .t/ C e t R1 gn .X t /; 0  t  1; .5:1:5/ is a martingale with respect to .F t ; P /;  2 S00 . By Doob’s inequality4 P



 1 An .1/  e Al .1//2 / sup jMn .t/  Ml .t/j > "  2 E ..e " 0t 1

which, combined with (5.1.4), means that there exists a subsequence ¹nk º such that P .Mnk .t / converges uniformly on Œ0; 1// D 1;

8 2 S00 :

.5:1:6/

In view of (5.1.5), (5.1.6), Theorem 2.2.3 and Lemma 5.1.2, we conclude that, by e , where selecting a new subsequence if necessary, Px .ƒ/ D 1; x 2 X n N eWe ƒ D ¹! 2  Ank .1; !/ < 1; e Ank .t; !/ converges uniformly in t on each finite interval of Œ0; 1/º5

.5:1:7/

e is some properly exceptional set containing N . and N Let us set e A.t; !/ D limnk !1 Re Ank .t; !/ for ! 2 ƒ and e A.t; !/ D 0 for t s e ! … ƒ. Furthermore, let A.t; !/ D 0 e d A.s; !/: A is then a PCAF with ƒ and e being its defining set and exceptional set respectively. N 4 Cf.

N. Ikeda and S. Watanabe [2]. e. the set defined by (A.2.22) for e S DX nN

5e  is

226

5 Stochastic analysis by additive functionals

In order to complete the proof, it suffices to show E .e A.1// D h; ui, 8 2 An .1/ is L2 .P /-convergent, S00 , by virtue of Theorem 2.2.3. Since Mn .1/ D e so is the martingale Mn .t/. Hence, A.t // C e t h; p t ui D lim E .Mn .t // D lim E .e An .1// E .e n!1

n!1

D lim h; R1 gn i D h; ui: n!1

By letting pt tend to infinity and noting the bound h; p t ui D E1 .U1 ; p t u/  p E1 ./ E1 .u; u/, we get the desired equality. Lemma 5.1.3. Consider  and A of the preceding Then forany ˛ > 0 R theorem. 1 and bounded non-negative Borel function f; Ex 0 e ˛t f .X t /dA t is a quasi continuous version of U˛ .f  /. Proof. It is sufficient to consider the case where ˛R D 1 and f D 1G ; G being 1 an open with .@G/ D 0. Let .x/ D Ex . 0 e t 1G .X t /dA t /; .x/ D R 1 set Ex . 0 e t 1X nG .X t /dA t /; then both  and are 1-excessive and  C D U1 . By Theorem 2.2.1 and Lemma 2.3.2,  and are quasi continuous versions of 1-potentials of some measures  and  2 S0 respectively. We know from Lemma 4.3.1 and the equalities  D HG1 ; D HX1 nG that suppŒ  G and suppŒ  X n G. Since  D  C , we have  D 1G  .

Theorem 5.1.2. For  2 S0 ; A 2 AC c of Theorem 5:1:1 is unique (up to equivalence). are associated with  2 S0 in the manner Proof. Suppose that A.1/ ; A.2/ 2 AC R 1 tc .1/ R1 .2/ of Theorem 5.1.1. Then Ex . 0 e dA t / D Ex . 0 e t dA t / D u.x/; x 2 X n N , for some properly exceptional set N . In the same way as in the proof of Theorem 5.1.1, Z 1  Z 1 .i / .j / e t dA t e t dA t vij .x/ D Ex 0

D

X

Z

Ex

kDi;j

0

0

1

e 2t u.X t /dA t

.k/

 ;

x 2 X n N; i; j D 1; 2. Let un D u ^ n. Then, by Lemma 5.1.3, Z 1  X .k/ E e 2t un .X t /dA t h; vij i D lim n!1

kDi;j

0

D lim 2h; U2 .un /i D 2hU2   u; i < 1 n!1

5.1

Positive continuous additive functionals and smooth measures

227

for any  2 S00 . Hence, ² Z

1

e

E

t

0

.1/ dA t 

Z

1

e

t

0

.2/ dA t

³2  D h; v11 2v12 Cv22 i D 0;  2 S00 ;

from which it easily follows that A.1/ is equivalent to A.2/ . Hereafter we use the notations Z Z t f .Xs /dAs ; UA˛ f .x/ D Ex .fA/ t D 0

1

e ˛t f .X t /dA t



0

C for A 2 AC c and f 2 B .

Lemma 5.1.4. Let  2 S0 and A 2 AC c be related as in Theorem 5:1:1. Then for any nearly Borel set B  X  Z B  Z .˛C /t ˛Ehm e f .X t /dA t " h.x/f .x/.dx/; ˛ " 1; X nB r

0

where h is any -excessive function .  0/; f 2 B C and B r is the set of regular points of B, i.e., B r D ¹x 2 X W Px .B D 0/ D 1º. Proof. It suffices to give the proof when h is a bounded -excessive function belonging to L2 .XI m/ and f 2 BbC . By Lemma 5.1.3, UA˛ f is a quasi continuous version of U˛ .f/. By virtue of Theorem 4.3.1, Z B  ˛t e f .X t /dA t D ˛.h; UA˛ f  HB˛ UA˛ f / ˛Ehm 0

D ˛E˛ .R˛ h; UA˛ f  HB˛ UA˛ f / D ˛E˛ .R˛ h  HB˛ R˛ h; UA˛ f / Z X nB R˛X nB h.x/f .x/.dx/ D ˛E˛ .R˛ h; U˛ .f// D ˛ X

which increases to

R X nB r

h.x/f .x/.dx/ as ˛ " 1. Here

R˛X nB h.x/

Z D Ex

B

e

˛t

 h.X t /dt :

0

C Lemma 5.1.5. Let A 2 AC c and f 2 Bb .X/.

(i)

ˇ

ˇ

UA˛ f  UA f C .˛  ˇ/R˛ UA f D 0;

˛; ˇ > 0:

228

5 Stochastic analysis by additive functionals

(ii) We put Z R˛A f .x/

D Ex

1

e

˛t A t

e

 f .X t /dt ;

0

x 2 X n N;

where N is a properly exceptional set for A. Then R˛A f is quasi continuous and R˛A f  R˛ f C UA˛ R˛A f D 0:

.5:1:8/

Proof. (i) is easy. (ii) was shown already in (4.6.21) in the special case that A t D Rt 0 g.Xs /ds.R The present case can be proved in the same way by using the identity t e A t  1 D 0 e As dAs and noting the next easily verifiable fact: R˛ f  R˛A f is ˛-excessive and consequently R˛A f is, together with R˛ f , finely continuous on X n N . In particular R˛A f is quasi continuous by virtue of Theorem 4.6.1. Before formulating the main assertions of this section, we need one more lemma. Recall the definition of the smooth measure given in §2.2:  2 S if  is a positive Borel measure charging no set of zero capacity and admits an associated generalized nest. Thus the class S contains all positive Radon measures charging no set of zero capacity. Lemma 5.1.6. Let ¹Fn º be an increasing sequence of closed sets. ¹Fn º is a generalized nest if and only if   .5:1:9/ Px lim X nFn <  D 0 q.e. x 2 X: n!1

Proof. Condition (2.2.17) for ¹Fn º to be a generalized nest is equivalent to   .5:1:10/ Px lim GnFn D 1 D 1 q.e. n!1

for any relatively compact open set G on account of Theorem 4.2.1. (5.1.9) implies (5.1.10) because of the quasi-left-continuity of the Hunt process M. To get (5.1.9) from (5.1.10), it suffices to choose a sequence ¹Gl º of relatively compact open sets with Gl  GlC1 ; Gl " X, and observe the inequality X nFn  Gl nFn ^ X nGl . Theorem 5.1.3. For  2 S and A 2 AC c , the following conditions are equivalent to each other: (i) For any -excessive function h (  0) and f 2 B C , 1 lim Ehm ..fA/ t / D hf  ; hi: t #0 t

.5:1:11/

5.1

Positive continuous additive functionals and smooth measures

229

(ii) For any -excessive function h (  0) and f 2 B C , ˛C

˛.h; UA

f / " hf  ; hi;

(iii) For any t > 0; f; h 2 B C , Ehm ..fA/ t / D

Z 0

˛ " 1:

.5:1:12/

t

hf  ; ps hids:

.5:1:13/

(iv) For any ˛ > 0; f; h 2 B C , .h; UA˛ f / D hf  ; R˛ hi:

.5:1:14/

When  2 S0 , each of the above four conditions is also equivalent to each of the following three conditions: (v) UA1 1 is a quasi continuous version of U1 . (vi) For any h 2 B C \ F and f 2 BbC , 1 hi: lim Ehm ..fA/ t / D hf  ; e t #0 t

.5:1:15/

(vii) For any h 2 B C \ F and f 2 BbC , hi: lim ˛.h; UA˛ f / D hf  ; e

˛!1

.5:1:16/

A PCAF A related to a given  2 S by .5:1:11/ is unique up to the equivalence. Conversely, given A 2 AC c , a smooth measure  satisfying .5:1:11/ is unique. 1 ˛t Ehm ..fA/ t / increases as t deProof. Let  2 S and A 2 AC c . Note that t e creases if h and f are as in (i). Therefore it suffices to prove the equivalence of (i)– (iv) when the right-hand sides are finite, for instance, when h 2 BbC , f D g  1Fn , g 2 BbC , ¹Fn º being a generalized nest associated with . Then, the uniqueness of the Laplace transform yields the implication (iv))(iii). (iii))(i))(ii) are obvious. (ii))(iv) is a consequence of Lemma 5.1.5 (i). Suppose next that  2 S0 and A 2 AC c . The implications (iii))(vi) and (iv))(vii) are clear. Lemma 5.1.3 means the equivalence (iv),(v). Let us derive the implication (vi))(v). Suppose that (vi) is satisfied and put c t .x/ D Ex .A t /. Then Z t Z Z 1 t 1 t h; ps hids D lim .ps h; cu /ds D lim .h; cuCs  cs /ds u#0 u 0 u#0 u 0 0 Z Z 1 t Cu 1 u .h; cs /ds  lim .h; cs /ds D .h; c t / D lim u#0 u t u#0 u 0

230

5 Stochastic analysis by additive functionals

for any -excessive function h in F . By taking the Laplace transform we get (iv) for h of this type and f D 1, which is enough to obtain (v). In the same way, we can get the implication (vii))(v). Suppose that A 2 AC c is related to a given  2 S by the condition (i). Take a generalized nest ¹Fn º such that 1Fn   2 S0 ; n D 1; 2; : : :, in accordance with Theorem 2.2.4. Then 1Fn  A and 1Fn   are related by (i) and consequently by (v). Hence 1Fn  A is unique by virtue of Theorem 5.1.2. Therefore A is uniquely determined by  on account of Lemma 5.1.6. The last statement in the theorem is trivial. We say that A 2 AC c and  2 S are in the Revuz correspondence if they satisfy the relation (5.1.11). In this case,  is called the Revuz measure of A and sometimes denoted by A . Theorem 5.1.4. The family of all equivalence classes of AC c and the family S are in one to one correspondence under the Revuz correspondence. The proof of Theorem 5.1.4 is accomplished by the following two lemmas and the uniqueness statements in Theorem 5.1.3. Lemma 5.1.7. Any A 2 AC c admits its Revuz measure  2 S . Proof. For a given A 2 AC c with a properly exceptional set N  X, we put .x/ D R1A f .x/; x 2 X n N , where f is a bounded Borel function in L2 .XI m/ such that f .x/ > 0; 8x 2 X. Then, .x/ > 0; 8x 2 X n N . Since  is quasi continuous by Lemma 5.1.5, there exists a sequence ¹En º of increasing closed sets T such that Cap.X n En / # 0; n ! 1; N  1 .X n En / and jEn is continuous nD1 for each n. Let us put ² ³ 1 Fn D x 2 En W .x/  .5:1:17/ n and prove that ¹Fn º satisfies condition (5.1.9). To this end, set Bn D ¹x 2 X n N W .x/  n1 º, n D Bn and  D limn!1 n . Since  is finely continuous on X n N , we have for x 2 X n N ,  Z 1 1 t A t e f .X t /e dt D Ex .e  n e An .X n //  : Ex n

n By letting n tend to infinity, we can see Px . < / D 0; x 2 X n N , in view of the strict positivity of f . Hence ¹Fn º satisfies (5.1.9) because of the inclusion X n Fn  .X n En / [ Bn and Theorem 4.2.1.

5.1

Positive continuous additive functionals and smooth measures

231

Now let An D 1Fn  A. Then UA1n 1 is 1-excessive (hence quasi continuous by Theorem 4.6.1) and satisfies the inequality UA1n 1  nUA1 '  nR1 f .2 F / by Lemma 5.1.5. Therefore we can use Theorem 2.2.1 and Lemma 2.3.2 to conclude that there exists a unique n 2 S0 such that UA1n 1 is a quasi continuous version of the potential U1 n . But then n D 1Fn  l ;

n 0; n D all A.n/ such that, for t  any ! 2 ƒ; A t .!/ D .1Fn  A 1; 2; : : : ; and .!/ D limn!1 X nFn .!/  .!/. For ! 2 ƒ let ´ .n/ A t .!/ D A t .!/;

X nFn1 .!/  t < X nFn .!/;

A t .!/ D A .!/ .!/;

t  .!/: .n/

Obviously A 2 AC c . Since A t D A t Z

XnFn

e

˛Ehm 0

.˛C /t

n D 1; 2; : : :

for t < X nFn , we see from Lemma 5.1.4,

 f .X t /dA t

Z "

.0/

Fn

h.x/f .x/.dx/;

˛ " 1;

232

5 Stochastic analysis by additive functionals

S S .0/ .0/ Fn being the fine interior. Note that the set n Fn n n Fn is exceptional because of (5.1.9). By letting n tend to infinity, we get ˛C

lim ˛.h; UA

˛!1

f / D hf  ; hi;

proving that  is the Revuz measure of A. The proof of Theorem 5.1.4 is now completed. Here we state two lemmas on A 2 AC c which will play important roles in the subsequent sections. The first one concerns an estimate of A, while the second one does its behaviors on a finely open subset. Lemma 5.1.9. For any A 2 AC c ;  2 S00 and t > 0, E .A t /  .1 C t/ kU1 k1  A .X/ . 1/:

.5:1:18/

Proof. (i) Assume first that the Revuz measure  D A of A is of finite energy integral:  2 S0 . By setting c t .x/ D Ex .A t /; x 2 X, we claim that ct 2 F

and

E.c t ; v/ D h;e v  p te v i;

8v 2 F :

.5:1:19/

By Theorem 5.1.3, the potential UA1 1.x/ of A is a quasi continuous version of the potential U1  of . Hence, h; c t i  e t h; UA1 1i D e t E1 ./ < 1. On the other hand, .c t ; c t  ps c t / D .c t ; cs  p t cs / D .c t  p t c t ; cs / which is, by Theorem 5.1.3, equal to Z s Z s h; pu .c t  p t c t /idu D h; 2c t Cu  cu  c2t Cu idu: 0

0

Therefore, we get 1 lim .c t ; c t  ps c t / D h; 2c t  c2t i D h; c t  p t c t i < 1; s#0 s proving that c t 2 F and E.c t ; c t / D h; c t  p t c t i. Similarly, we have the relation E.c t ; v/ D h;e v  p te v i; 8v 2 F . For  2 S0 and  2 S00 , (5.1.19) gives us

e

e

E .A t / D h; c t i D E1 .U1 ; c t / D h; U1   p t .U1 /i C .c t ; U1 /   Z c t .x/m.dx/ ;  kU1 k1 .X/ C X

5.1

Positive continuous additive functionals and smooth measures

233

which proves the inequality (5.1.18) by noting that Z 1 1 c t .x/m.dx/ D lim Em .A t / D .X/: sup t #0 t t >0 t X When  D A is a general smooth measure, we may consider the increasing sequence ¹Fn º of closed sets of Theorem 2.2.4.   Since 1Fn  A has the Revuz measure 1Fn   2 S0 , we have E .1Fn  A/ t  .1 C t / kU1 k1  .Fn /. By letting n tend to infinity, we arrive at (5.1.18). C For a nearly Borel finely open set G, A 2 AC c and f 2 B , we use the notations Z 1 G f .x/ D E .f .X / W t <  /; R f .x/ D e ˛t p G pG x t X nG t ˛ t f .x/dt; 0 Z XnG  Z 1  G;˛ ˛t ˛t e f .X t /dA t D Ex e f .X t /dA t ^ XnG : UA f .x/ D Ex 0

0

Lemma 5.1.10. Let A 2 AC c and G be a nearly Borel finely open set. C (i) If h is -excessive (  0) on G with respect to ¹p G t º and f 2 B , then G;˛C

˛.h; UA

f /m " hf 1G  A ; hi;

˛ " 1:

(ii) For any t > 0; f; h 2 B C Z t  Z t Ehm f .Xs /dAs^ XnG D hf 1G  A ; psG hids: 0

0

(iii) Suppose that Px .A t D 0; 8t < X nG / D 1

.5:1:20/

for m-a.e. x 2 X. Then A .G/ D 0 and .5:1:20/ holds for q.e. x 2 X. Proof. (i) When A 2 S0 , this has been proved in Lemma 5.1.4 and indeed the proof of Lemma 5.1.4 works for any -excessive function h with respect to ¹p G t º rather than ¹p t º. When A is a general smooth measure, we choose closed sets ¹Fn º as in the proof of the previous lemma. We may then let n ! 1 in the relation (i) holding for 1Fn  A and 1Fn  A . (ii) We have G;ˇ

UAG;˛ f  UA

G;ˇ

f C .˛  ˇ/R˛G UA

f D 0;

˛; ˇ > 0;

an equation analogous to Lemma 5.1.5 (i). We then see the equivalence of the statements (i) and (ii) exactly in the same manner as in the proof of Theorem 5.1.3. (iii) (5.1.20) holding m-a.e. x 2 X implies 1G  A D 0 by (ii) and 1G  A D 0 by Theorem 5.1.4.

234

5 Stochastic analysis by additive functionals

Let us study the relationship between the support of PCAF and that of the Revuz measure. For A 2 AC c with a properly exceptional set N , we let e D ¹x 2 X n N W Px .R D 0/ D 1º Y

.5:1:21/

e is called a support of A. We consider where R.!/ D inf¹t > 0 W A t .!/ > 0º. Y R eD the function 'A .x/ D Ex .e /. 'A is easily seen to be 1-excessive. Since Y e is nearly Borel, finely closed with respect to MX nN ¹x 2 X n N W 'A .x/ D 1º, Y e is actually a and quasi closed in virtue of Theorem 4.6.1. We shall prove that Y quasi support (in the sense of §4.6) of the Revuz measure of A. e. For this, we need a lemma stating that A t .!/ increases only when X t .!/ 2 Y e Denote by  the first hitting time of Y . Lemma 5.1.11. For q.e. x 2 X, Px .R D / D 1;

(5.1.22)

Px .A t D .1 e Y  A/ t ; 8t > 0/ D 1:

(5.1.23)

e Px -a.s. by Lemma A.2.7 and accordingly Proof. For x 2 X n N; X 2 Y Px . < R/ D P . < R; R ı  > 0/ D Ex .PX .R > 0/I  < R/ D 0: We next show Px .R < / D 0 q.e. x 2 X:

.5:1:24/

e and t > 0 In fact we have for any x 2 .X n N / n Y Px .R < / D Px .ARCt > 0; R < / D Ex .PXR .A t > 0/; R < / e/;  Ex .PXR .R < t/; XR 2 .X n N / n Y er , it holds which tends to zero as t # 0. Since (5.1.24) is trivially true for x 2 Y r e e except for x 2 N [ .Y n Y / and hence for q.e. x by Theorem 4.1.3. The proof of (5.1.22) is complete. We next introduce the random time sets I.!/ D ¹t W A t C" .!/  A t .!/ > 0; 8" > 0º; eº; Z.!/ D ¹t W X t .!/ 2 Y and show that Px .I  Z/ D 1 q.e. x 2 X:

.5:1:25/

5.1

Positive continuous additive functionals and smooth measures

235

Indeed, by virtue of the right continuity of 'A .X t /, we have  [ Px .I 6 Z/  Px ¹t 2 I; 'A .X t / < 1º t

X



Px .Aq  Ar > 0; r C  ı r  q/;

r 0 W As .!/ > t º; then Z

1

e

ˇEx 0

ˇ t

 vn .X t / dt

Z D ˇEx

1

e 0

ˇA t

 vn .X t /dA t

D 0 q.e. x 2 X:

Since vn is q.e. finely continuous by Theorem 4.2.2, we get after letting ˇ ! 1 e and hence v D 0 q.e. on Y e. that vn D 0 q.e. on Y Finally, we turn to the study of PCAF’s in the strict sense. An AF in the strict sense is by definition an AF admitting a defining set ƒ with Px .ƒ/ D 1; 8x 2 X,

236

5 Stochastic analysis by additive functionals .1/

namely, an AF without exceptional set N . Two such functionals A t regarded to be equivalent if .1/

Px .A t

.2/

D A t / D 1;

8t > 0; 8x 2 X:

.2/

and A t

are

.5:1:26/

We can then find a common defining set ƒ with Px .ƒ/ D 1; 8x 2 X, such that .1/ .2/ A t .!/ D A t .!/; 8t  0; 8! 2 ƒ. We denote by AC c;1 the family of all PCAF’s in the strict sense. In the rest of this section, we assume that M satisfies the absolute continuity condition (4.2.9). Recall a consequence of this assumption stated at the end of §4.2: the ˛-potential U˛  R of  2 S0 admits a quasi continuous and ˛-excessive modification R˛ .x/ D X r˛ .x; y/.dy/ where r˛ .x; y/ is the resolvent density appearing in Lemma 4.2.4. In characterizing the family AC c;1 analytically, the subfamily S00 of S0 defined by (2.2.10) plays a primary role.  2 S00 if and only if  is a positive Borel measure such that .X/ < 1 and R1 .x/ is uniformly bounded in x 2 X. We first prove the following as a direct consequence of Theorem 5.1.1. Theorem 5.1.6. For any  2 S00 , there exists A 2 AC c;1 uniquely such that Z Ex

1

e t dA t



0

D R1 .x/;

8x 2 X:

.5:1:27/

Proof. Since  2 S0 , Theorem 5.1.1 provides us with a finite A 2 AC c , with a defining set ƒ and an exceptional set N such that (5.1.27) holds for m-a.e. x 2 X. Since both sides are 1-excessive with respect to MjX nN , (5.1.27) holds for any x 2 X n N . We shall construct a desired e A 2 AC c;1 by modifying A. We set M D supx2X R1 .x/ .< 1/. We may assume that P .ƒ/ D 1 and P .! 2  W A t .!/ D 0; 8t > 0/ D 1. Let A"tT .!/ D A t " ." !/, t > " > 0, ! 2 , and, for a sequence "n # 0, ƒ0 D n "1 ƒ. Then Px .ƒ0 / D 1; 8x 2 X, because n Px ."1 ƒ/ n

Z D Ex .PX"n .ƒ// D

X

p"n .x; dy/Py .ƒ/ C Px .  "n /P .ƒ/

which is equal to 1 by the assumption (4.2.9). If we set Ant .!/ D A"t n .!/; ! 2 , then for each ! 2 ƒ0 Ant .!/  Am t .!/ D A"m "n ."n !/;

m < n;

5.1

Positive continuous additive functionals and smooth measures

237

which means that Ant .!/ is increasing as n ! 1 and the convergence is locally uniformly in t 2 .0; 1/ if the limit is finite. On the other hand, from Z 1 Z 1 t s t e dAs  e e s dAs At  e 0

t

holding on ƒ, we have Ex .A t /  e t R1 .x/  p t R1 .x/;

x 2 X n N;

and consequently Ex .A"t / D Ex .EX" .A t " //  e t " p" R1 .x/  p t R1 .x/;

8x 2 X:

Put, for ! 2  and t > 0, ´ limn!1 Ant .!/ if the limit exists e A t .!/ D 0 otherwise: e A t is then an adapted process and further A t /  lim Ex .Ant /  e t R1 .x/  p t R1 .x/  e t M; Ex .e n!1

A0C / D lim Ex .e At /  0 Ex .e t #0

for every x 2 X. Therefore we can conclude that e A 2 AC c;1 with defining set e D ¹! 2 ƒ0 W e A t .!/ < 1 for any t > 0; e A0C .!/ D 0º: ƒ R R t s n t e we see that e dAs D e t Ant C "n e s Ans ds increases, as For ! 2 ƒ, " n Rt n ! 1, to 0 e s d e As . Hence, for any x 2 X, Z 1  Z 1  s e s n Ex e d As D lim Ex e dAs n!1

0

"n

 Z "n D lim Ex e EX"n n!1

1

e

s

 dAs

0

D lim p"n R1 .x/ D R1 .x/; n!1

getting (5.1.27) for e A. To prove the uniqueness, take A.1/ ; A.2/ 2 AC c;1 satisfying (5.1.27). We already know by Theorem 5.1.2 that (5.1.26) holds for q.e. x 2 X. Hence, for any x 2 X, .1/

Px .A t

.2/

 A.1/ "n D A t

It now suffices to let n ! 1.

.1/

.2/

 A.2/ "n / D Ex .PXs .A t "n D A t "n // D 1:

238

5 Stochastic analysis by additive functionals

A positive Borel measure  on X is said to be smooth in the strict sense if there exists a sequence ¹En º of Borel sets increasing to X such that 1En   2 S00 for each n and   Px lim X nEn   D 1; 8x 2 X: .5:1:28/ n!1

The totality of the smooth measures in the strict sense is denoted by S1 . Theorem 5.1.7. (i) The equivalence class of AC c;1 is in one to one correspondence with the class S1 under the Revuz correspondence .5:1:11/. (ii) S1  S . (iii)  2 S1 if and only if there exists a sequence ¹En º of Borel finely open sets increasing to X such that 1En   2 S00 for each n. Proof. (i) The uniqueness of the correspondence follows from the following observation and Theorem 5.1.6: A 2 AC c;1 and  2 S00 are under the Revuz correspondence if and only if the equation (5.1.27) holds. In fact, by virtue of Theorem 5.1.3, the Revuz correspondence is equivalent to (5.1.27) holding for m-a.e. x 2 X, and consequently for every x 2 X because both sides of (5.1.27) are 1-excessive on X. The construction of A 2 AC c;1 for a given  2 S1 can be done analogously to the proof of Lemma 5.1.8 with the help of Theorem 5.1.6. The construction of  2 S1 for a given A 2 AC c;1 is also similar to the proof of Lemma 5.1.7, but this time we may set ² ³ 1 En D x 2 Gn W .x/ > .5:1:29/ n where Gn are relatively compact open sets increasing to X. Note that En is then Borel and finely open because  D R1A f is a difference of 1-excessive functions by Lemma 5.1.5 and hence Borel measurable in view of assumption (4.2.9) and Theorem A.2.5. C (ii) This follows from (i) and the inclusion AC c;1  Ac . (iii) If ¹En º satisfies the stated condition, then (5.1.28) is valid on account of the quasi-left-continuity of M. If  2 S1 , then we can take by (i) the associated A 2 AC c;1 , which in turn enables us to redefine ¹En º by (5.1.29). Example 5.1.1. Let M D .; M; X t ; Px / be the Brownian motion on Rd . The associated Dirichlet form E on L2 .Rd / is given by . 12 D; H 1 .Rd // (see Example 4.2.1). The absolutely continuous measure .dx/ D f .x/dx;

f  0;

5.1

Positive continuous additive functionals and smooth measures

239

is smooth with respect to the form E when f is locally integrable. In fact,  is then a positive Radon measure charging no set of zero capacity since  vanishes on any set of Lebesgue measure zero. The PCAF of M with Revuz measure  is Z t f .Xs .!//ds: .5:1:30/ A t .!/ D 0

If f is locally bounded, then R1 .1Br f /.x/  supx2Br f .x/ < 1 and consequently  2 S1 . (5.1.30) then gives an associated PCAF in the strict sense. Similar assertions hold even when  diverges around a set of zero capacity. Suppose d  2. Then, for example, f .x/ D jxjˇ ;

x 2 Rd ;

gives a smooth measure for any real ˇ. Indeed, Fn D ¹x 2 Rd W 1=n  jxj  nº serves as a generalized nest because the one point set ¹0º is of zero capacity (see Example 2.2.2 and Example 2.2.3). In view of (2.2.27), we see that  2 S1 if d  3 and ˇ > 2. The associated PCAF A t is still given by (5.1.30). A t satisfies the finiteness condition  Z t ˇ Px jXs j ds < 1 D 1; t > 0; .5:1:31/ 0

whenever x ¤ 0, since ¹0º is polar (see Example 4.2.1). Because of the law of the iterated logarithm, however, (5.1.31) is violated for x D 0 when ˇ  2. In this case, A t is not a PCAF in the strict sense. But it is always a PCAF in our sense because we can ignore the polar set ¹0º as an exceptional set for A t . When d D 1, the Brownian motion on R1 admits no non-empty exceptional set (Example 4.5.1). Hence, the present definition of AF reduces to the strict one. By virtue of the remark at the end of Example 2.1.2, we easily see that a measure  on R1 is smooth if and only if  is a positive Radon measure. In particular, the ı-measure ı¹yº concentrated at one-point y admits a PCAF 2L.t; y; !/ according to Theorem 5.1.4. L.t; y/ is called the Brownian local time. It is known that there exists a set ƒ   with Px .ƒ/ D 1; 8x 2 R1 , such that for each ! 2 ƒ, L.t; y; !/ is continuous in two variables t and y.6 Notice that the formula (5.1.13) serves as a characterization of the correspondence of  2 S and A 2 AC c . By integrating the corresponding formula for  D ı¹yº and A t D 2L.t; y/ by 1E dy, we get Z t Z 1E .Xs /ds D 2 L.t; y/dy; E  R1 : .5:1:32/ 0

6 Cf.

N. Ikeda and S. Watanabe [2].

E

240

5 Stochastic analysis by additive functionals

Furthermore, we get the expression A t .!/ D 2

Z L.t; y; !/.dy/

.5:1:33/

R1

for the PCAF A with a Revuz measure being a positive Radon measure  on R1 . The quasi support of  coincides with the topological support suppŒ on account of Theorem 4.6.2. Hence the support of A equals suppŒ by Theorem 5.1.5. Example 5.1.2. We continue to consider the d -dimensional Brownian motion. Let d  3. We exhibit a measure  2 S00 whose topological support and quasi support differ by a set of positive capacity. Let Fn be the set of points in Rd defined by Fn D ¹z D .x1 ; : : : ; xd / W xk D 2n .1 C 2l/; l 2 N; k D 1; 2; : : : ; d º: S S Put Dn D z2Fn Brn .z/ for rn D 2n.d C1/=.d 2/ . Then D D 1 nD1 Dn is an d open set dense in R . Consider a smooth measure  2 S00 such that .Rd / < 1, .Rd n D/ D 0, .Brn .z// > 0, 8z 2 Fn ; n  1. Note that such measure exists. Indeed, by enumerating the points of Fn as Fn D ¹zn;k ºk1 , it suffices to set 1 X

D

2nk 1Brn .zn;k /  m

.5:1:34/

n;kD1

where m denotes the Lebesgue measure. Since .Rd / is finite and the Newtonian .0/ potential of  is bounded,  2 S00  S00 (see Example 2.2.1). Z t X nk 2 1Brn .zn;k / .Xs /ds At D n;k

0

e) is the associated PCAF in the strict sense of the Brownian motion. Let Y (resp.Y be the topological support (resp. quasi support) of . Then, by virtue of Theoe D ¹z 2 Rd W 'A .z/ D 1º where 'A .z/ D Ez .e R / and R.!/ rem 5.1.5 Y D inf¹t > 0 W A t .!/ > 0º. e/ > 0, which implies that Obviously Y D Rd . We shall show that m.Rd n Y d e e Cap.Y n Y / > 0. It suffices to prove that R n Y D ¹z W 'A .z/ < 1º contains at e, then least one point, for, if there exists a point z0 2 Rd n Y e t p t 'A .z0 / " 'A .z0 / < 1;

t # 0:

e/ > 0, because otherwise This implies that m.Rd n Y e t p t 'A .z0 / D e t p t 1.z0 / ! 1 as t ! 0 by the absolute continuity of p t .z0 ;  / relative to m.

241

5.2 Decomposition of additive functionals of finite energy

Since A increases only when the Brownian motion stays in D r , to show that e, it suffices to prove Pz0 .D D 0/ D 0. We shall prove this for z0 2 Rd n Y z0 D 0 by using Wiener’s test:7 P0 .D D 0/ D 0 if and only if

1 X

2k.d 2/ Cap.Bk / < 1;

kD1

where Bk D D \ .B2k .0/ n B2k1 .0//. Since the function ur .x/ D .jxj2d = r 2d / ^ 1 belongs to H 1 .Rd / and ur .x/  1 on Br .0/ for any r > 0, Cap.0/ .Br .0// 

1 D.ur ; ur / D C.d /r d 2 2

for some constant C.d / depending on d . Noting that the number of points of z 2 Fn for which Brn .z/ intersects with B2k .0/ does not exceed .2nk C 2/d , we have Cap .Dn \ B2k .0//  .2nk C 2/d Cap .Brn .0//  C.d /.2nk C 2/d 2n.d C1/  C 0 .d /2kd 2n e by for some constant C 0 .d /. We can see now that 0 … Y 1 X

2k.d 2/ Cap .D \ B2k .0// 

kD1

1 X

2k.d 2/ Cap .Dn \ B2k .0//

k;nD1

 C 0 .d /

1 X

22k < 1:

kD1

5.2

Decomposition of additive functionals of finite energy

From now on till the end of Chapter 5, we shall deal with those AF’s of the process M which are not necessarily non-negative. For any AF A t of M, we set e.A/ D lim t #0

1 Em .A2t / 2t

.5:2:1/

whenever the limit exists. e.A/ is called the energy of A. First of all, we shall exhibit three important classes of AF’s of finite energy. 7 Cf.

K. Itô and H. P. McKean [1].

242

5 Stochastic analysis by additive functionals

(I) AF’s generated by functions Suppose that a function u on X possesses a version e u .e u D u m-a.e.) such that e u is finely continuous q.e. and finite q.e. Then Œu

At

De u.X t /  e u.X0 /;

t >0

.5:2:2/

defines a finite AF in our sense and, indeed, a unique equivalence class independent of the choice of the version e u by virtue of Lemma 4.1.5. u 2 Fe a quasi AŒu is well-defined for any u 2 Fe because we may take as e Œu continuous version of u (Theorem 2.1.7 and Theorem 4.2.2). Moreover, A t is of finite energy and Z 1 Œu res .5:2:3/ e.A / D E .u; u/ C e u.x/2 k.dx/  E.u; u/; u 2 Fe ; 2 according to the formula (4.5.22). Here E res is the resurrected Dirichlet form defined by (4.5.20). E res equals E if and only if there is no killing inside X. If u 2 F or u 2 Fe in transient case, AŒu is cadlag in view of Lemma 4.2.2 and Corollary 4.3.1. (II) Martingale AF’s of finite energy Consider the family M D ¹M W M is a finite cadlag AF such that for each t > 0 Ex .M t2 / < 1 and Ex .M t / D 0 q.e. x 2 Xº:

.5:2:4/

Since Em .M t2 / is subadditive in t , e.M / is well-defined and 1 Em .M t2 / . 1/ t >0 2t

e.M / D sup

.5:2:5/

for any M 2 M. We set MV D ¹M 2 M W e.M / < 1º:

.5:2:6/

M 2 M if and only if M is a square integrable perfect AF in the strict sense with mean zero of the Hunt process MjX nN , N being some properly exceptional set (depending on M in general). In other words, M belongs to the space Mad with respect to the process MjX nN in the sense of Appendix A.3. Hence we can conclude from the following Theorem A.3.18 that M admits its sharp bracket hM i

5.2 Decomposition of additive functionals of finite energy

243

as a PCAF in the strict sense of MjX nN . Thus each M 2 M admits a PCAF hM i 2 AC c such that Ex .hM i t / D Ex .M t2 / q.e. x 2 X; t > 0:

.5:2:7/

Such hM i is unique for M up to the equivalence in the sense of §5.1. We call M 2 M a martingale additive functional (MAF in abbreviation). hM i is called its quadratic variation or the sharp bracket associated with M . Denote by hM i the Revuz measure of hM i according to Theorem 5.1.3. hM i is called the energy measure of the MAF M . From (5.2.1), (5.2.7) and (5.1.11), we see that the energy of an MAF is just a half of the total mass of its energy measure: 1 e.M / D hM i .X/; 2

M 2 M:

.5:2:8/

For M; L 2 M let hM; Li t D

1 ¹hM C Li t  hM i t  hLi t º : 2

.5:2:9/

Then hM; Li t is a CAF of bounded variation on each finite interval of t and satisfies Ex .M t L t / D Ex .hM; Li t /; 8t > 0; q.e. x 2 X: .5:2:10/ Furthermore the signed measure hM;Li defined by 1 hM;Li D ¹hM CLi  hM i  hLi º 2

.5:2:11/

is related to hM; Li by the relation (5.1.11). Since hM;N i is bilinear, symmetric with respect to M; L 2 M and hM;M i D hM i  0 for all M 2 M, we get for all non-negative f 2 L2 .XI hM i C hLi / sZ ˇZ ˇ sZ ˇ ˇ ˇ f dhM;Li ˇˇ  f dhM i  f dhLi ; ˇ X X X sZ ˇ sZ ˇsZ ˇ ˇ ˇ f dhM i  f dhLi ˇˇ  f dhM Li : (5.2.12) ˇ X

X

X

(III) CAF’s of zero energy Nc D ¹N W N is a finite continuous AF; e.N / D 0; Ex .jN t j/ < 1 q.e. for each t > 0º:

.5:2:13/

244

5 Stochastic analysis by additive functionals

The quadratic variation of N 2 Nc vanishes in the following sense: ŒnT X

.N.kC1/=n  Nk=n /2 ! 0;

n ! 1; in L1 .Pm /

.5:2:14/

kD1

because the expectation of the left-hand side equals ŒnT X

2 2 Em .EXk=n .N1=n //  nT  Em .N1=n / ! 0;

n ! 1:

kD1

An example of N 2 Nc is given by Z Nt D

t

f .Xs /ds

.5:2:15/

0

for any nearly Borel function f 2 L2 .XI m/. Then Z

t

Ex 0

 jf .Xs /jds  e t R1 jf j.x/

is finite q.e. by Theorem 4.2.3 and N is a CAF. Furthermore, Z Em .N t2 /

D 2Em

Z

t

f .Xs / 0

Z t Z D 2Em

0

f .Xv /dvds s



t s

f .Xs /pv f .Xs /dvds 0 0 t s

Z tZ D2



t

0

.ps 1; f  pv f /dvds:

Hence 1 1 Em .N t2 / D 2t t

Z 0

Z

t

.p t s 1; f  Ss f /ds;

Ss f .x/ D

Since the right-hand side of (5.2.16) is dominated by (5.2.15) is of zero energy, i.e., N 2 Nc .

s

pv f .x/dv: .5:2:16/ 0 1 t

Rt 0

sds  .f; f /, N of

5.2 Decomposition of additive functionals of finite energy

245

e

Another example is the PCAF A with Revuz measure  belonging to S0 . Then Ex .A t /  e t U1 .x/ < 1 q.e. and Z t  2 Em .A t / D 2Em .A t  As /dAs 0

D 2 lim Em

n X

n!1

 .A t  Ak t =n /.Ak t =n  A.k1/t =n /

kD1

D 2 lim Em

n X

n!1

 .EXkt=n .A t k t =n //.Ak t =n  A.k1/t =n /

kD1

 2 lim Em

n X

n!1

kD1

Z D 2Em

 .EXkt=n .A t //.Ak t =n  A.k1/t =n / 

t 0

 2t h; E:.A t /i

EXs .A t /dAs

by (5.1.13), and further h; E:.A t /i  e t h; UA1 1  e t p t UA1 1i ! 0;

t ! 0:

We are particularly interested in the sum of the classes (II) and (III): A D MV ˚ Nc

.5:2:17/

namely, A consists of AF’s A such that At D Mt C Nt ;

V N 2 Nc : M 2 M;

.5:2:18/

Clearly A is a linear space of AF’s of finite energy. Moreover, the expression (5.2.18) of A 2 A is unique because MV \ Nc D ¹0º where 0 denotes the additive functional identically zero. In fact, if A 2 MV is of zero energy, then hAi vanishes by (5.2.8) and so does hAi. Hence, A D 0 by (5.2.7). We define the mutual energy of A; B 2 A by e.A; B/ D lim t #0

1 Em .A t B t /: 2t

.5:2:19/

We know by Schwarz inequality that e.A; B/ D 0 when either A or B is in Nc . Therefore, e.A/ D e.M /

if

A D M C N;

V N 2 Nc : M 2 M;

.5:2:20/

246

5 Stochastic analysis by additive functionals

The main purpose of this section is to show that any AF AŒu .u 2 Fe / in the first class (I) actually belongs to A, i.e., AŒu is expressed as a sum of elements of MV and Nc uniquely. To this end we first prove that the space MV is a Hilbert space with inner product (5.2.19). Theorem 5.2.1. MV is a real Hilbert space with inner product e. More specifiV there exists a unique M 2 MV and cally, for any e-Cauchy sequence M .n/ 2 M, .n/ a subsequence nk such that limn!1 e.M  M / D 0 and for q.e. x 2 X,   .n / Px lim M t k D M t uniformly on any finite interval of t D 1: k!1

Proof. By virtue of Lemma 5.1.9 and identities (5.2.7), (5.2.8), we have E .M t2 /  2.1 C t/kU1 k1 e.M /;

 2 S00 :

.5:2:21/

.M t ; P / t 0 is then a square integrable martingale and Doob’s inequality implies P



 2 sup jMs j > ˛  2 .1 C T /kU1 k1 e.M /: ˛ 0sT

.5:2:22/

Assume that M .n/ 2 MV constitutes an e-Cauchy sequence and select a subsequence nk ! 1 such that e.M .nkC1 /  M .nk / / < 1=23k . By (5.2.22),   1 2.1 C T /kU1 k1 .n / P : sup jMs kC1  Ms.nk / j > k  2 2k 0sT Applying the Borel–Cantelli lemma to this inequality, we see P .ƒc0 / D 0; .n /

 2 S00

.5:2:23/

where ƒ0 D ¹! 2  W Ms k converges uniformly in s on each finite intervalº. In view of Theorem 2.2.3, (5.2.23) implies T Px .ƒ0 / D 1 q.e. x 2 X. Denote by .n / k and put ƒ1 D k k . Furthermore, we set Ms .!/ D k a defining set of M .n / limk!1 Ms k .!/; s  0, for ! 2 ƒ0 \ ƒ1 . Ms is then an AF with defining set .n/ ƒ0 \ ƒ1 . Since M t is L2 .P / convergent to M t by virtue of (5.2.21), we see E .M t2 / < 1 and E .M t / D 0 for any  2 S00 . It is easy to conclude from this with the help of Theorem 2.2.3 that Ex .M t2 / < 1 and Ex .M t / D 0 q.e. x 2 X, that is, M 2 M. For any " > 0, choose N such that e.M .n/  M .m/ / < ", .n/ n; m > N . We have .1=2t/Em ..M t  M t /2 /  " by Fatou’s lemma and (5.2.5). Then e.M .n/  M /  " by (5.2.5) again. Hence, M 2 MV and M .n/ is e-convergent to M .

5.2 Decomposition of additive functionals of finite energy

247

Theorem 5.2.2. For any u 2 Fe , the AF AŒu can be expressed uniquely as AŒu D M Œu C N Œu ;

V M Œu 2 M;

N Œu 2 Nc :

.5:2:24/

Moreover, it holds that e.M

Œu

1 / D E .u; u/ C 2 res

Z e u.x/2 k.dx/  E.u; u/:

.5:2:25/

Proof. We already know the uniqueness of the decomposition (5.2.24) and it suffices to show the existence of such M Œu and N Œu . We start with the special case where u is in the range of the resolvent: u D R1 f , f being a Borel function in L2 .XI m/. In this case (5.2.24) is reduced to the usual semimartingale decomposition, namely, we may set ´ Rt Œu N t D 0 .u.Xs /  f .Xs // ds .5:2:26/ Œu Œu M t D u.X t /  u.X0 /  N t ; t  0: As we have seen, N Œu of (5.2.26) belongs to Nc . Since u is an element of F , Œu A t D u.X t /u.X0 / is of finite energy and M Œu of (5.2.26) satisfies the relation V (5.2.25) in view of (5.2.3). We easily see that M Œu 2 M and hence M Œu 2 M. Next, take any Borel function u 2 F and define the approximating function un by un D nRnC1 u D R1 fn ; fn D n.u  nRnC1 u/: .5:2:27/ By the uniqueness of the decomposition (5.2.24) for un ’s, we have then M Œun   M Œum  D M Œun um  and Z 1 Œun  Œum  res .un  um /2 d k: .5:2:28/ M / D E .un  um ; un  um / C e.M 2 R Since un 2 F and E1 -convergent to u and E.v; v/ dominates E res .v; v/C12 v 2 d k V for any v 2 F , we conclude that M Œun  is an e-Cauchy sequence in the space M. By virtue of Theorem 5.2.1, the formula ´ V e/ M Œu D limn!1 M Œun  in .M; .5:2:29/ N Œu D AŒu  M Œu ; u 2 F makes sense and M

Œu

V 2 M;

e.M

Œu

1 / D E .u; u/ C 2 res

Z e u.x/2 k.dx/;

u2F:

.5:2:30/

248

5 Stochastic analysis by additive functionals

It only remains to show that N Œu of (5.2.29) belongs to the space Nc . Note that a subsequence nk exists and Px .N Œunk  converges to N Œu uniformly on any finite interval of t / D 1 q.e. x 2 X; (5.2.31) because the same statements for AŒun  and M Œun  hold on account of Lemma 5.1.2 and Theorem 5.2.1 respectively. From this and (5.2.26), we know that N Œu is a CAF. On the other hand, we have from (5.2.29) that Œu

Nt

Œuun 

D At

Œu

 .M t

Œun 

 Mt

Œun 

/ C Nt

and, consequently lim t #0

1 Œu Em ..N t /2 /  3e.AŒuun  / C 3e.M Œu  M Œun  /: 2t

By virtue of (5.2.3), (5.2.28) and (5.2.29), the right-hand side is dominated by 6E.u  un ; u  un /, which can be made arbitrarily small with large n. Therefore, e.N Œu / D 0 and N Œu 2 Nc . Finally suppose that u 2 Fe . Let ¹un º  F be an approximating E-Cauchy sequence such that un ! u m-a.e. In view of (5.2.28) and Theorem 5.2.1, Œu  limn!1 M t n converges uniformly on any finite interval Px -a.s. Therefore, similarly to the previous case, it suffices to prove the uniform convergence of ¹AŒun  º on any finite interval. If M is transient, this follows from Lemma 5.1.2. In the general case, by virtue of Lemma 1.6.7, there exists a bounded strictly positive function g 2 K such that u 2 Feg . Then the approximating sequence ¹un º can be taken in such a way that ¹un º is contained in F g and Cauchy relative to E g . Owing to Lemma 4.6.7, we can take the canonical subprocess .X tg ;  g ; Px / of M by the multiplicative Rt functional exp. 0 g.Xs /ds/ as a Hunt process associated with .E g ; F g /. Since .E g ; F g / is transient, Lemma 5.1.2 implies that Px .ƒgT / D 0 q.e., where ® ƒgT D ! 2  W lim

¯ sup je un .Xsg /  e u.Xsg /j > 0 :

n!1 0sT

Therefore, for ® ƒT D ! 2  W lim

¯ sup je un .Xs /  e u.Xs /j > 0 ;

n!1 0sT

5.2 Decomposition of additive functionals of finite energy

249

RT

Px .ƒT /  e kgk1 T Ex .e  0 g.Xs /ds I ƒT /   Z T g kgk1 T Px ƒT I g.Xs /ds < Z D 0 q.e. x 2 X; De 0

as was to be proved for general u 2 Fe .8 Corollary 5.2.1. The following linearity and continuity hold: (i) M ŒauCbv D aM Œu C bM Œv ; N ŒauCbv D aN Œu C bN Œv ; a; b 2 R1 , u; v 2 Fe . (ii) un , u 2 F (2 Fe in transient case), E1 .un  u; un  u/ ! 0; n ! 1 Œunk 

) there exists a subsequence ¹unk º such that limnk !1 M t Œunk 

limnk !1 N t q.e. x 2 X.

Œu

D Nt

Œu

D Mt ;

uniformly on each finite interval of t , Px -a.s. for

Proof. The relation (i) follows from the uniqueness of the decomposition (5.2.24). The equality (5.2.25) means that the linear transform ˆ W u 7! M Œu

.5:2:32/

V e/. Combinis continuous from the Dirichlet space .Fe ; E/ into the space .M; ing this with Theorem 5.2.1 and Lemma 5.1.2, we conclude the continuity statement (ii). (5.2.24) also implies the following embedding statement telling us that the structure of the Dirichlet space may be studied entirely within the framework of the space MV of martingales of finite energy. Corollary 5.2.2. Suppose that the process M has no killing inside X. Then the transformation ˆ of .5:2:32/ is an isometry and the extended Dirichlet space V e/. .Fe ; E/ can be identified by ˆ with a closed subspace of .M; Having established decomposition (5.2.24), we are now in a position to study some basic properties of AFs M Œu and N Œu in (5.2.24). In this section we shall give basic formulae of computing M Œu and N Œu . First of all, we identify the energy measure of the martingale part M Œu with the energy measure hui of the function u in the sense of §3.2. Denote by Fb and Fe;b the family of all 8Z

is an exponentially distributed random variable independent of X t (see A.2).

250

5 Stochastic analysis by additive functionals

m-essentially bounded functions in F and Fe , respectively. We note that Fe;b is an algebra (Corollary 1.6.3) and further that 1 .u  p t u; v/ ! E.u; v/; t

t # 0; u 2 Fe ; v 2 F :

.5:2:33/

Indeed u  p t u 2 L2 .XI m/ even when u 2 Fe by virtue of Lemma 1.5.4, and hence the left-hand side of (5.2.33) is finite and equal to the polarized expression of the left-hand side of the formula (4.5.7). Therefore we get (5.2.33) from (4.5.7). Keeping this in mind, let us proceed to the proof of Theorem 5.2.3. hM Œu i equals hui for u 2 Fe . In particular, for u 2 Fe;b , Z e.x/ Œu .dx/ D 2E.u  f; u/  E.u2 ; f /; f 2 Fb : .5:2:34/ f hM i X

Proof. We first show (5.2.34) for u 2 Fe;b . By (5.1.11) and (5.2.7), Z e.x/ Œu .dx/ D lim 1 Ef m ..M Œu /2 / f t hM i t #0 t X

.5:2:35/

Œu

when f is a bounded -excessive function .  0/ in F . Since M t D e u.X t /  Œu Œu 2 Œu 2 e u.X0 /  N t and Ef m ..N t / /  kf k1 Em ..N t / /, we can see that the right-hand side of (5.2.35) equals ³ ² 2 1 1 2 2 2 lim Ef m ..e u.X t /  e u.X0 // / D lim .uf; u  p t u/  .f; u  p t .u // t t #0 t t #0 t which is equal to the right-hand side of (5.2.34) because u; u2 2 Fe ; f 2 F ; uf 2 Fe \ L2 D F and (5.2.33) applies. e. Suppose E is Next take any f 2 Fb . (5.2.34) is now valid for fn D nRn f transient, then fn and ufn are E-weakly convergent to f and uf respectively on account of Corollary 1.5.1. Since fn converges boundedly to f q.e. as well, we get (5.2.34) for f from that for fn . In the general case, we choose g 2 K so that u 2 Feg . Since g 2 L1 .XI m/, the same E-weak convergences as above take place on Feg and we can get (5.2.34) in the same way. (5.2.34) along with (3.2.14) leads us to the identity hM Œu i D hui for u 2 Fb . This identity readily extends to u 2 Fe by noting the inequality (5.2.12) and the bound ˇ ˇZ ˇ ˇ ˇ f dhM Œu i ˇˇ  2kf k1 E.u; u/ ˇ X

which follows from (5.2.8) and (5.2.25).

5.2 Decomposition of additive functionals of finite energy

251

We next state a characterization of N Œu . Theorem 5.2.4. The following conditions are equivalent for an AF A and for a function u 2 Fe : (i) A D N Œu , (ii) A 2 Nc and, for each t > 0, Ex .A t / D p t u.x/  u.x/; (iii) A 2 Nc and

m-a.e. x 2 X;

1 lim Evm .A t / D E.u; v/; t #0 t

8v 2 F :

.5:2:36/

.5:2:37/

(iv) A 2 Nc and .5:2:37/ holds for v D R1 f for any bounded Borel f 2 L1 .XI m/. Proof. The implication (i) ) (ii) is clear. (ii) ) (iii) follows from the equation (5.2.33). (iii) ) (iv) is trivial. Assume that an AF A satisfies the condition (iv). Let us put c t .x/ D Ex .A t /, then we see from A 2 Nc and the relation c t Cs .x/ D c t .x/ C p t cs .x/ q.e. for each t; s > 0;

.5:2:38/

that c t is in L2 .XI m/ and L2 -right continuous in t > 0. By (5.2.37), (5.2.38), (1.5.5) and Lemma 1.5.4, we have that for any v of the stated type, 1 1 .v  pT v; c t / D lim .ST v  ps ST v; c t / D lim .ST v  p t ST v; cs / s#0 s s#0 s D  E.ST v  p t ST v; u/ D  E.ST v; u  p t u/ D  .v  pT v; u  p t u/;

t > 0; T > 0:

Therefore, l t D .v; c t C u  p t u/ satisfies the equation l t D l t CT  lT of the linearity. Since lim t #0 .1=t/l t D 0 by (5.2.37) and (5.2.33), l t vanishes and we get (5.2.36). Finally, we assume condition (ii) for A. In order to derive (i), we set B t D Œu Œu u.X t /  e u.X0 /  M t  A t ; q t .x/ D Ex .B t2 / and claim that Nt  At D e q t .x/ D 0 q.e. for each t > 0:

.5:2:39/

Clearly B t is a CAF of zero energy R and Ex .B t / D 0 for m-a.e. x 2 X on account of the assumption (ii). But then X q t .x/m.dx/ D 0 because the left-hand side

252

5 Stochastic analysis by additive functionals

is subadditive in t . We thus have q t .x/ D 0 m-a.e., from which we can conclude ps q t D 0 q.e. for each s > 0. By using Fatou’s lemma, q t .x/ D Ex



 lim .B t C1=n  B1=n /2  lim p1=n q t .x/ D 0 q.e.;

n!1

n!1

getting (5.2.39). So far in this section, we have treated AF’s of various types but always admitting exceptional sets. Under the extra condition of the absolute continuity of the transition function, we may get a decomposition like (5.2.24) involving only AF’s in the strict sense as will be suggested by the next theorem. We call M t .!/ a MAF in the strict sense if it is a finite cadlag AF in the strict sense and .5:2:40/ Ex .M t / D 0; Ex .M t2 / < 1; 8x 2 X: According to Theorem A.3.18, there exists then a unique hM i 2 AC c;1 such that Ex .hM i t / D Ex .M t2 /;

8x 2 X:

We call hM i the sharp bracket of M in the strict sense. Theorem 5.2.5. Assume that M satisfies the absolute continuity condition .4:2:9/. Suppose that a function u satisfies the following conditions: (i) u 2 F ; u is bounded and finely continuous. (ii) hui 2 S00 . (iii) 9 D  .1/   .2/ with  .1/ ;  .2/ 2 S00 and, for a dense subset L of F , E.u; v/ D h;e v i;

8v 2 L:

Then Œu

u.X t /  u.X0 / D M t

Œu

C Nt

Px -a.s.; 8x 2 X:

.5:2:41/

Œu

Here M t is an MAF in the strict sense whose sharp bracket in the strict sense has the Revuz measure hui . Further N Œu D A.1/ C A.2/

Px -a.s.; 8x 2 X;

.5:2:42/

where A.1/ (resp. A.2/ ) is a PCAF in the strict sense with Revuz measure  .1/ (resp.  .2/ ).

5.2 Decomposition of additive functionals of finite energy

253

Proof. The existence of the PCAF A.i / in the strict sense with Revuz measure  .i / has been assured by Theorem 5.1.7, i D 1; 2. By virtue of Theorem 5.1.3 and condition (iii), A D A.1/ C A.2/ satisfies the equation (5.2.37) for any v 2 F and consequently A is a strict version of N Œu according to Theorem 5.2.4. We define N Œu by (5.2.42) and set Œu

Mt

Œu

D u.X t /  u.X0 /  N t ;

t > 0:

Œu V In particular Ex .M Œu / D 0 Then M t is an AF in the strict sense and M Œu 2 M. t .i / q.e. By noting that Ex .A t / < 1; 8x 2 X; i D 1; 2, we get Œu

Œu

Ex .M t C" /  Ex .M"Œu / D Ex .EX" .M t // Z Œu p" .x; dy/Ey .M t / D 0; D X

8x 2 X;

and by letting " # 0, we are led to Œu

Ex .M t / D 0;

8x 2 X:

Denote by B the PCAF in the strict sense with Revuz measure hui (Theorem 5.1.7). Then by Theorem 5.2.3 Œu

Ex ..M t /2 / D Ex .B t /

q.e.

Therefore as in the above Œu

Ex ..M t C"  M"Œu /2 / D Ex .B t C"  B" /;

8x 2 X

.5:2:43/

and by Fatou’s lemma Œu

Ex ..M t /2 /  Ex .B t /;

8x 2 X:

.5:2:44/

The right-hand side is dominated by e t R1 hui .x/ < 1. We have proved that M Œu is an MAF in the strict sense. Since the sharp bracket in the strict sense of M Œu has hui as its Revuz measure by Theorem 5.2.3, it coincides with B by virtue of Theorem 5.1.7, and consequently the equality takes place in (5.2.44). But we derive the equality more directly for the sake of later use. Œu Denoting M t by M t , we then have for any x 2 X q ˇ q ˇq ˇ 2 ˇ 2 ˇ Ex ..M t C"  M" / /  Ex .M t /ˇ  Ex ..M t C"  M t  M" /2 / q q q 2 2 D Ex ..M" ı  t  M" / /  Ex .EX t .M" // C Ex .M"2 / p p  Ex .EX t .B" // C Ex .B" /;

254

5 Stochastic analysis by additive functionals

which tends to 0 as " # 0. Hence, by letting " # 0 in (5.2.43), we can replace the inequality in (5.2.44) by the equality as was desired. Example 5.2.1. Consider the case where X is an Euclidean domain D and M is an m-symmetric diffusion process on D whose Dirichlet form E on L2 .DI m/ possesses C01 .D/ as its core. We assume for simplicity that M admits no killing inside D. From Theorem 3.2.3 and Theorem 4.5.3, we know then that E should have the expression E.u; v/ D

d Z X i;j D1 D

@u.x/ @v.x/ ij .dx/; @xi @xj

u; v 2 C01 .D/;

with some measures ij satisfying (1.2.3). Now the right-hand side of (5.2.34) equals 2

d Z X i;j D1 D

@u.x/ @u.x/ f .x/ij .dx/ for u; f 2 C01 .D/; @xi @xj

and thus (5.2.34) implies hM Œu i .dx/ D 2

d X @u.x/ @u.x/ ij .dx/; @xi @xj

u 2 C01 .D/;

.5:2:45/

i;j D1

which equals the expression (3.2.30) of hui of course. Suppose that every ij is absolutely continuous with respect to the Lebesgue P measure. Take Borel density functions aij .x/ such that dij D1 aij .x/i j  0, 8x 2 D, 8 2 Rd . In view of Theorem 5.1.3, the identity (5.2.45) then implies hM

Œu

Z t X d  @u @u it D 2 aij .Xs /ds; @xi @xj 0

u 2 C01 .D/:

.5:2:46/

i;j D1

Furthermore, if aij satisfies 0 jj2 

d X

aij .x/i j  ƒ0 jj2 ;

8x 2 D; 8 2 Rd ;

.5:2:47/

i;j D1

for some positive constants 0 and ƒ0 and if m.dx/ D dx, then (5.2.46) holds for any u 2 Fb namely, for any bounded function in H01 .D/, because so does (5.2.45).

5.2 Decomposition of additive functionals of finite energy

255

Example 5.2.2 (Skorohod representation of the reflecting Brownian motion). Let M D .X t ; Px / be the reflecting Brownian motion on a Lipschitz domain D  Rd considered in Example 4.5.3. Denote by m the Lebesgue measure on D. M is an m-symmetric conservative diffusion on D satisfying the absolute continuity condition (4.2.9) and possessing the regular Dirichlet form .E; F / D . 12 D; H 1 .D// on L2 .D/. C01 .D/ is actually a core of E. The surface measure  on the boundary @D can be defined as follows. Consider a coordinate neighbourhood D \ U D ¹.x 0 ; xd / W xd > F .x 0 /º \ U with a Lipschitz function F . Since F is differentiable a.e. with bounded rF , we may let for a Borel B  @D \ U Z .1 C jrF .x 0 /j2 /1=2 dx 0 ; B D ¹x 0 W .x 0 ; F .x 0 // 2 Bº: .B/ D B

-a.e. point x 2 @D then admits a unit inward normal vector n.x/ D .n1 .x/; : : : ; nd .x// and an associated divergence theorem holds. Consider the coordinate functions 'i .x/ D xi ; 1  i  d , and let '.x/ D .'1 .x/; : : : ; 'd .x// ; x 2 D. Then 'i 2 F and d X @'i @'j  m D ıij m; @xk @xk kD1 Z Z 1 1 @v dx D  e v .x/ni .x/.dx/; v 2 C01 .D/; E.'i ; v/ D 2 D @xi 2 @D

h'i ;'j i D

(5.2.48) (5.2.49)

due to the divergence theorem. Let r˛ .x; y/; ˛ > 0; x; y 2 D, be the R resolvent density of M in accordance with Lemma 4.2.4. Then R1 m.x/ D D r1 .x; y/m.dy/ D 1, x 2 D, and so m 2 S00 . Using a known bound9 r1 .x; y/  C jx  yj.d 2/ R ; d > 2 ; r1 .x; y/  1 C.log jxyj _ 1/; d D 2, we also have that R1 .x/ D @D r1 .x; y/.dy/ is bounded and hence  2 S00 . Therefore, in virtue of Theorem 5.1.7, there are unique PCAF’s in the strict sense with Revuz measure m and  respectively. The former coincides with the constant AF t and the R t latter denoted by L t is called the local time of M on the boundary. Obviously 0 ni .Xs /dLs is then a CAF in the strict sense with (signed) Revuz measure ni  . Theorem 5.2.5 now applies in getting Œ'

'.X t / D '.X0 / C M t 9 Cf.

R. F. Bass and P. Hsu [2].

Œ'

C Nt

Px -a.s.; 8x 2 D

256

5 Stochastic analysis by additive functionals Œ'

where M t is a continuous MAF in the strict sense with hM Œ'i  ; M Œ'j  i t D ıij  t Rt Œ'  and N t i D 12 0 ni .Xs /dLs . Using a martingale characterization of Brownian motion10 , we arrive at the representation Z 1 t n.Xs /dLs ; Px -a.s.; 8x 2 D; .5:2:50/ X t D X0 C B t C 2 0 where B t denotes a d -dimensional Brownian motion with respect to the filtration of M. Notice that the third term in the right-hand side represents a singular inward drift at the boundary which forces X t to stay inside D against the isotropic nature of the Brownian motion B t . We call (5.2.50) a Skorohod representation of the reflecting Brownian motion X t . Theorem 5.2.2, Theorem 5.2.3 and Theorem 5.2.5 will be extended in §5.5 when E is local.

5.3

Martingale additive functionals and Beurling–Deny formulae

In §3.2, we derived the Beurling–Deny formulae in an analytical way. In this section we give a probabilistic approach to them by making use of the decomposition Œu

u.X0 / D M t e u.X t /  e

Œu

C Nt ;

V N Œu 2 Nc ; u 2 F ; M Œu 2 M;

in Theorem 5.2.2 combined with the general theory of martingale additive functionals summarized in Appendix A.3. The latter will enable us to give a further decomposition of the martingale part M Œu . Let us first make the following observation. As was stated in the beginning of 5:1, a functional B of M is an AF in the present sense if and only if B is an AF in the strict sense of the Hunt process MjX nN , N being a properly exceptional set depending on B in general. Therefore Theorems A.3.17, A.3.18, A.3.19 of Appendix A.3 still work in producing AF’s in the present sense. We shall take this for granted in the sequel. In particular, Theorem A.3.18 and the paragraph following it provide us, for a given M 2 M, with its square bracket Œ M  and sharp bracket hM i as PAF’s. Actually hM i is a PCAF. We have made this observation for hM i in the last section already. Theorem A.3.18 further implies that M 2 M is uniquely decomposed as M D Mc C Md; 10 Cf.

N. Ikeda and S. Watanabe [2].

M c 2 Mc ; M d 2 Md ;

5.3

Martingale additive functionals and Beurling–Deny formulae

257

where Mc D ¹M 2 M W Px .M t is continuous in t / D 1; q.e. xº;

(5.3.1)

Md D ¹M 2 M W hM; Li D 0; Px -a.e. for q.e. x for any L 2 Mc º: MAF’s M c and M d are called the continuous part and the purely discontinuous part of the MAF M respectively. Let us denote by M Œu;c and M Œu;d the continuous part and purely discontinuous part of the MAF M Œu for u 2 F . We further let M t D M t  M t  , t > 0, Œu M 2 M. Since N t is continuous and e u./ D 0, we have that Œu

u.X  /1¹ t º : M 1¹ t º D e By virtue of the quasi-left continuity of the Hunt process, we see that there exists a CAF .e u.X  /1¹ t º /p such that u.X  /1¹ t º  .e u.X  /1¹ t º /p M Œu;k D e is a purely discontinuous MAF. In fact, we may take as .e u.X  /1¹ t º /p the dual predictable projection of e u.X  /1¹ t º in accordance with Lemma A.3.1 and Theorem A.3.17. Put M Œu;j D M Œu;d  M Œu;k : Then the MAF M Œu is decomposed as M Œu D M Œu;c C M Œu;j C M Œu;k :

(5.3.2)

Note that hM i is the dual predictable projection Œ M p of Œ M  and Œ M  t D P hM c ; M c i t C 0 0 W X t … Kº and v is any function expressible as Z 1 K e ˛t p K pK v.x/ D R˛ f .x/ D t f .x/dt; t f .x/ D Ex .f .X t /I t < K / 0

for a bounded Borel function f . Proof. We have ŒEvm .N t I t  K /2  kvk1 Em .N t2 /Pjvjm .t  K / which is o.t 2 /, because N is of zero energy and Z 1 1 K K Pjvjm .t  K /  .R˛ jf j; 1  p t 1/ ! .jf j  ˛R˛K jf j/d m; t t t #0 e K e being the fine interior of K. K Lemma 5.4.4. Let  D  .1/   .2/ be a difference of finite Borel measures on X charging no set of zero capacity. If u 2 Fe satisfies E.u; v/ D h;e v i;

8v 2 Fb;F

.5:4:17/

for some closed set F , then Px .N Œu D A.1/ C A.2/ on Œ0; F // D 1;

q.e. x 2 X;

.5:4:18/

where A.i / is a PCAF with Revuz measure  .i / , i D 1; 2. Proof. Under the assumption,  D  .1/  .2/ for some smooth measures  .i / with  .i / .X/ < 1; i D 1; 2. First suppose that u 2 F and u satisfies equation (5.4.17). We can choose a generalized nest ¹Fk º such that 1Fk  .i / 2 S0 ; i D 1; 2, for each b k D Fk \ F and b k by virtue of Theorem 2.2.4. We put F k D 1b  . C u  m/. Fk Then (5.4.17) implies k ;e v i; E1 .u; v/ D hb Owing to Theorem 4.3.1, we have

8v 2 Fb;b : F k

A

u; v/ D E1 .U1b k  H 1 c .U1b k /; v/; E1 .u  H 1 c e b b Fk Fk

268

5 Stochastic analysis by additive functionals

which can be readily extended to v 2 Fb . Hence Fk

A

k C H 1 c e u  H 1 c .U1b k /: u D U1b b b Fk Fk

.5:4:19/

We can conclude from (5.4.19) combined with Corollary 5.2.1 (i), Lemma 5.4.1 and Lemma 5.4.2 that (5.4.18) holds with F being replaced by b . By letting Fk k ! 1, we arrive at (5.4.18) on account of Lemma 5.1.6. Suppose next that E is transient, u 2 Fe and u satisfies the equation (5.4.17). By using the 0-order version of Theorem 2.2.4, we can find a generalized nest .0/ ¹Fk º such that 1Fk   .i / 2 S0 ; i D 1; 2, for each k (see Exercise 2.2.4). If we b k D Fk \ F and b  k D 1b  , then in the same way as above we get from put F Fk (5.4.17) and Theorem 4.3.2 that

e

E.u  Hb e u; v/ D E.Ub  k  Hb .Ub  k /; v/; Fc Fc k

k

8v 2 Fb;b : F k

For any bounded v 2 Fe;b , we can find a sequence of uniformly bounded wn 2 F Fk E-convergent to v as n ! 1. Then vn D wn  Hb w f 2 Fb;b is E-convergent Fc n F k

k

to v  Hb e v D v. Therefore the above identity holds for any bounded v 2 Fe;b Fc F

k

k

and eventually for any v 2 Fe;b . Thus we get F k

e

e u  Hb .Ub  k /; u D Ub  k C Hb Fc Fc k

k

which leads us to (5.4.18) by the same reasoning as before. Finally we consider the general case that u belongs to a general (not necessarily transient) extended Dirichlet space Fe and satisfies the equation (5.4.17). We use g Lemma 1.6.7 to find g 2 K with u 2 Fe . If we put  D  C ug  m, then  is still a finite signed Borel measure charging no set of zero capacity and moreover E g .u; v/ D h;e v i;

8v 2 Fb;F :

Since E g is transient and F g D F , we conclude from what was just proved that   Z  g g;Œu .1/ .2/ D A C A  u  g.Xs /ds on Œ0; F / D 1 q.e. x 2 X; Px N 0

.5:4:20/ with respect to the corresponding canonical subprocess Mg D .X tg ;  g ; Px / employed already in the proof of Theorem 5.4.1. We then get (5.4.18) from (5.4.20) just as in the proof of Theorem 5.4.1.

5.4 Continuous additive functionals of zero energy Œu

Proof of Theorem 5:4:2. Suppose N t Œu

Nt

.1/

D A t

.2/

C At

269

is of bounded variation on Œ0; /, then Px -a.s. on ¹t < º q.e. x;

.1/ and  .2/ the Revuz measures of A.1/ for some A.1/ ; A.2/ 2 AC c . Denote by  .2/ and A respectively. Let ¹Fk º be a generalized compact nest commonly associated with  .1/ and  .2/ . Then ¹Fk º is also associated with the smooth signed measure  D  .1/   .2/ . We may assume that IFk   .i / 2 S0 ; i D 1; 2, for every k. Fix k and denote the notions of Lemma 5.4.3 relevant to Fk by k and R˛k f . Since 1Fk  A.i / 2 Nc , i D 1; 2, we get from Lemma 5.4.3 and Theorem 5.1.3 that

1 1 Œu Œu lim Evm .N t / D lim Evm .N t I t < k / t #0 t t #0 t 1 D lim Evm ..1Fk  A.1/ / t C .1Fk  A.2/ / t I t < k / t #0 t 1 D lim Evm ..1Fk  A.1/ / t C .1Fk  A.2/ / t / D h; vi t #0 t for v D R˛k f with bounded Borel f . Hence (5.4.16) holds for this type of function v in view of Theorem 5.2.4. For k a general v 2 Fb;Fk , vn D nRnC1 v are of the above type, uniformly bounded, convergent to v and E1 -convergent to v as well. Thus we attain (5.4.16). The converse implication is immediate from Lemma 5.4.4 and Lemma 5.1.6. The following corollary extends Lemma 5.4.1. Corollary 5.4.1. Suppose u 2 F satisfies E.u; v/ D h; vi;

8v 2 C;

.5:4:21/

where C is a special standard core of E and  D 1  2 a difference of positive Radon measures charging no set of zero capacity. Then N Œu D A.1/ C A.2/ where A.i / is a PCAF with Revuz measure i ; i D 1; 2. S In fact, for relatively compact open sets ¹Gk º with G k  GkC1 , 1 kD1 Gk D X, any v 2 Fb;GN k  FGkC1 admits a uniformly bounded sequence vn 2 CGkC1 , n D 1; 2; : : : ; converging to e v q.e. and in E-metric as well by virtue of Theorem 4.4.3. Hence (5.4.21) implies (5.4.16) for ¹Fk º D ¹G k º. In connection with Theorem 5.4.2, we like to know when a finite signed measure charges no set of zero capacity.

270

5 Stochastic analysis by additive functionals

Theorem 5.4.3. Assume that the resolvent R˛ of M satisfies the following condition: .5:4:22/ R˛ .Bb .X//  Cb .X/; ˛ > 0: Suppose that  is a finite signed measure on X and, for some u 2 Fe , E.u; v/ D h; vi;

8v 2 F \ Cb .X/:

.5:4:23/

Then  charges no set of zero capacity. Proof. (5.4.23) holds for any bounded 1-excessive function v in F because it holds for R˛ v on account of the assumption (5.4.22) and further ˛R˛C1 v is increasing to v and E-convergent to v as ˛ ! 1. Let K be a compact set of zero capacity. Since (5.4.22) implies that R˛ .x;  / is absolutely continuous with respect to m and consequently K is polar on account of Theorem 4.1.2; Px .K < 1/ D 0; 8x 2 X. T Choose a sequence of relatively compact open sets Gn such that GnC1 G n and 1 nD1 Gn D K. Then eGn .x/ D  

P G K n / decreases as n ! 1 to Ex .e Ex .e / D 1K .x/ by virtue of (A.2.9) and further E1 .eGn ; eGn / ! 0; n ! 1. Since (5.4.23) is valid for v D eGn , we get .K/ D 0 by letting n ! 1. Corollary 5.4.2. Assume that X is compact and that condition .5:4:22/ is fulfilled. If u 2 Fe satisfies jE.u; v/j  C kvk1 ;

8v 2 F \ Cb .X/;

.5:4:24/

for some constant C > 0, then N Œu is of bounded variation. Indeed (5.4.24) implies by the Riesz representation theorem that u satisfies the equation (5.4.23) for some finite signed measure . We can then get the desired conclusion from Theorem 5.4.2 and Theorem 5.4.3. The first two theorems of this section will be extended in the next section when E is local.

5.5

Extensions to additive functionals locally of finite energy

The decomposition AŒu D M Œu C N Œu of Theorem 5.2.2 can be extended to u in a broad class of functions including Floc under the hypothesis .E; F / possesses the strong local property:

.5:5:1/

5.5

Extensions to additive functionals locally of finite energy

271

From now on till the end of this section, we assume (5.5.1). According to Theorem 4.5.3, we may then assume that M is a diffusion process on X with no killing inside in the sense that (4.5.19) holds for every x 2 X. The killing measure k disappears in the Beurling–Deny formula for E. We can now utilize the next lemma. u1 D e u2 q.e. on a nearly Borel finely Lemma 5.5.1. (i) For any u1 ; u2 2 F , if e open set G, then Œu1 

Mt

Œu2 

D Mt

8t < G ; Px -a.s. for q.e. x 2 X;

;

.5:5:2/

where G D inf¹t > 0 W X t … Gº the first leaving time from G. (ii) M D Mc : any MAF is continuous. Proof. Let u D u1  u2 . Under (5.5.1), M Œu D M Œu;c in view of (5.3.2) Œu and Theorem 5.3.1. Lemma 5.3.1 then implies that M t D 0; t < G . Since M ŒR˛ g D M ŒR˛ g;c ; 8˛ > 0; 8g 2 C0 .X/, the purely discontinuous part of M 2 M vanishes on account of Theorem A.3.20. We notice that, by (4.5.19) holding for all x 2 X, Px .t ^ G < / D 1;

8t > 0; 8x 2 X;

.5:5:3/

provided that G is relatively compact. At this stage, it is convenient to introduce the notion of local AF’s. We call A t .!/ a local additive functional (local AF in abbreviation) if A t .!/ satisfies all requirements for an AF stated in the beginning of §5.1 except that the additivity A t Cs .!/ D As .!/ C A t .s !/ for ! 2 ƒ is required only for non-negative t; s with t C s < .!/. Therefore a local AF is a synonym for an AF in the case that M is conservative; Px . D 1/ D 1, 8x 2 X. For two local AF’s A.1/ ; A.2/ and a stopping time   , we write .1/

At

.2/

D At ;

t < ;

if Px -probability of this event equals Px .t < / for each t  0 and q.e. x 2 X. .1/ .2/ There is then a common defining set ƒ such that A t .!/ D A t .!/ for any ! 2 ƒ and any t < .!/. If  can be taken to be  in the above, then we say that A.1/ and A.2/ are equivalent and write A.1/ D A.2/ : Let us also introduce the family „ of sequences of finely open sets defined by „ D ¹¹Gn º W Gn is nearly Borel S finely open for all n, Gn  GnC1 ; 1 nD1 Gn D X q.e.º:

.5:5:4/

272

5 Stochastic analysis by additive functionals

We say that a function u is locally in F in the broad sense (u 2 FPloc in notation) if there exists a sequence ¹Gn º 2 „ and a sequence ¹un º  F such that u D un m-a.e. on Gn . Denote by FPb;loc the family of functions u 2 FPloc for which each function un of the sequence appearing in the definition of u can be chosen bounded. FPloc (resp. FPb;loc ) is obviously broader than the space Floc (resp. Fb;loc ) introduced at the end of §3.2. u. Lemma 5.5.2. (i) Any function u 2 FPloc has a quasi continuous modification e (ii) For ¹Gn º 2 „, Px



 lim Gn D  D 1 q:e: x 2 X:

n!1

(iii) For an AF A and a nearly Borel finely open set G, A.t Cs/^G D As^G C A t ^G ı s^G Px -a.s.;

8x 2 X n N;

where N is any properly exceptional set containing .X n G/ n .X n G/r and an exceptional set for A. Proof. (i) Let ¹Gn º 2 „ and ¹un º  F be associated with u 2 FPloc . Take quasi continuous modifications e un of un . Since ¹Gn º are quasi open by Theorem un jGn q.e., n D 1; 2; : : : ; in view of 4.6.1, u has a version e u such that e uj G n D e Lemma 2.1.5. e u is easily seen to be quasiScontinuous. (ii) We may assume that N D X n 1 nD1 Gn is properly exceptional. Suppose that Px . < / > 0 for some x 2 X n N where  denotes limn!1 Gn . Since XGn 2 .X n Gn / [  by Lemma A.2.7, we can conclude from the quasileft-continuity of M that Px . < ; X 2 N / > 0 contradicting the proper exceptionality of N . (iii) Since .X n G/ n .X n G/r is exceptional by Theorem 4.1.3, we have from Lemma A.2.7 that XG 2 .X n G/r [  Px -a.s., 8x 2 X n N if N is chosen as in the statement. Hence G ı G D 0 and the stated additivity of A is valid. The third statement of the above lemma will be used in the proof of Theorem 5.5.5 and Theorem 5.5.6. For any u 2 FPloc , its quasi continuous version e u is q.e. finite, q.e. finely continuous and satisfies the properties stated in Theorem 4.2.2. Hence Œu

At

De u.X t /  e u.X0 /;

t  0;

u./ D 0). defines an AF AŒu in the sense of §5.1 (we set e

5.5

Extensions to additive functionals locally of finite energy

273

By an MAF locally of finite energy, we mean a local AF M t admitting a sequence ¹Gn º 2 „ and a sequence of MAF’s ¹M .n/ º  MV such that, for each n, .n/

Mt D Mt

t < G n :

The family of MAF’s locally of finite energy is denoted by MV loc . By a CAF locally of zero energy, we mean a local AF N t admitting a sequence ¹Gn º 2 „ and a sequence ¹N .n/ º  Nc such that, for each n, .n/

Nt D Nt ;

t < Gn :

We denote by Nc;loc the family of all CAF’s locally of zero energy. The next theorem extends Theorem 5.2.2. Theorem 5.5.1. For any u 2 FPloc the AF AŒu can be decomposed as AŒu D M Œu C N Œu ;

ı

M Œu 2 Mloc ; N Œu 2 Nc;loc :

.5:5:5/

Such a decomposition is unique up to the equivalence of local AF’s. Proof. Lemma 5.5.1 readily yields the decomposition (5.5.5). Indeed, let ¹Gn º 2 „ and ¹un º  F be associated sequences of u. Let AŒun  D M Œun  CN Œun  be the Œu  Œu  corresponding decomposition given by Theorem 5.2.2. Since M t n D M t nC1 for t < Gn by Lemma 5.5.1, M Œu D limn!1 M Œun  ; t <  I M Œu D 0; t  , Œu Œu Œu is well-defined and belongs to MV loc . It then suffices to let N t D A t  M t : Before proving the uniqueness of the decomposition, we make the following observation. For a real valued process A let v kt .A/

D

Œk t  X

.A.lC1/=k  Al=k /2 ;

k D 1; 2; : : :

lD1

which approximates a quadratic variation of A. We further let A t D A t ^ for a V then M is a square integrable Pm -martingale and stopping time . If M 2 M, lim v kt .M / D hM i t ^

k!1

in L1 .Pm /

in view of Lemma 5.5.1 (ii) and Theorem A.3.10. On the other hand, for N 2 Nc , v kt .N / is L1 .Pm /-convergent to 0 by virtue of (5.2.14) and hence v kt .N / admits a subsequence converging to 0 Pm -a.s.

274

5 Stochastic analysis by additive functionals

We now assume that M t CN t D 0, M t 2 MV loc ; N t 2 Nc;loc . We can choose an associated sequence ¹Gn º 2 „ in common for M and N . We can assume that each Gn is relatively compact and we have by noticing (5.5.3) and Lemma 5.5.1 (ii) .n/

.n/

M t ^n C N t ^n D 0;

8t > 0; n D Gn ; n D 1; 2; : : : :

The preceding observation implies Em .hM .n/ i t ^n / D 0; 8t > 0; and consequently hM .n/ i t D 0; t < n , in view of Lemma 5.1.10 (iii). Hence M D 0 proving the uniqueness of the decomposition. Although Theorem 5.5.1 is formulated for the class FPloc , we shall be mainly concerned with the class Floc . For u 2 Floc , we can take as an associated sequence in „ any relatively compact open sets Gn with G n  GnC1 ;

1 [

n D 1; 2; : : : ;

Gn D X:

.5:5:6/

nD1

If u; v 2 Floc and u D v on a relatively compact open set G, then Œu

Mt

Œv

D Mt ;

Œu

Nt

Œv

D Nt ;

t < G :

.5:5:7/

The next theorem is immediate from Theorem 5.2.3. Note that, for M 2 MV loc , its quadratic variation hM i t , t  0, is well-defined as a PCAF. Theorem 5.5.2. For u 2 Floc , hM Œu i coincides with hui introduced at the end of §3.2. If u 2 Fb;loc , then the equation .5:2:34/ holds for any f 2 Fb with compact support. Two theorems concerning the support and bounded variation property of N Œu demonstrated in the preceding section can be readily extended to u 2 Floc as well. Note that the spectrum .u/; ˛ .u/ are well-defined for u 2 Floc in an obvious manner. Theorem 5.5.3. Theorem 5:4:1 remains valid for u 2 Floc if we replace F and F˛ in .5:4:6/ and .5:4:7/ by F ^  and F˛ ^  respectively. Proof. Consider relatively compact open sets ¹Gn º satisfying (5.5.6). For u 2 Floc , choose un 2 F with u D un on Gn for each n. If F D .u/, then E.un ; v/ D E.u; v/ D 0; 8v 2 F \ C0 .X/ with suppŒv  .X n F / \ Gn , and accordingly .un /  F [.X nGn /. Theorem 5.4.1 (i) yields (5.4.6) with u and F Œu Œu  being replaced by un and F [.X nGn / respectively. Since N t D N t n ; t < Gn , and Gn " , we let n ! 1 to arrive at (5.4.6) for u 2 Floc with F being replaced by F ^ . In the same way, we get the present version of (5.4.7).

5.5

275

Extensions to additive functionals locally of finite energy

Theorem 5.5.4. Theorem 5:4:2 remains valid for u 2 Floc if we take as ¹Fk º in the statement (ii) a generalized compact nest associated with . Proof. The implication (ii))(i) is clear from Lemma 5.4.4. To prove (i))(ii), we assume that N Œu for u 2 Floc is of bounded variation and consider an associated generalized nest ¹Fk º as in the proof of Theorem 5.4.2. Fix k and denote the notions of Lemma 5.4.3 relevant to K D Fk by k and R˛k f . Choose w 2 F such that w D u in a neighbourhood of Fk . Then, from Lemma 5.4.3, Theorem 5.2.4 and (5.5.7), we get for any v D R˛k f with bounded f 1 1 Œu Œw lim Evm .N t I t < k / D lim Evm .N t I t < k / t #0 t t #0 t 1 Œw D lim Evm .N t / D E.w; v/ D E.u; v/: t #0 t The rest is the same as the proof of Theorem 5.4.2. Corollary 5.5.1. Corollary 5:4:1 remains valid for u 2 Floc . We can now give an extension of Theorem 5.2.5. Theorem 5.5.5. Assume the absolute continuity condition .4:2:9/. Suppose that a function u satisfies the following conditions: (i) u 2 Fb;loc , u is finely continuous on X. (ii) 1G  hui 2 S00 for any relatively compact open set G. (iii) 9 D  .1/   .2/ with 1G   .1/ ; 1G   .2/ 2 S00 for any relatively compact open set G and E.u; v/ D h; vi; 8v 2 C; for some special standard core C of E. Let A.1/ ; A.2/ and B be PCAF’s in the strict sense with Revuz measures  .1/ ; .2/  and hui respectively in accordance with Theorem 5:1:7. Then Œu Œu u.X t /  u.X0 / D M t C N t ; Px -a.s. 8x 2 X: .5:5:8/ Here N Œu D A.1/ C A.2/ ;

Px -a.s. 8x 2 X;

.5:5:9/

M Œu

is a local AF in the strict sense such that, for any relatively compact and open set G, Œu

Ex .M t ^G / D 0; Œu

8x 2 G;

Ex ..M t ^G /2 / D Ex .B t ^G /;

(5.5.10) 8x 2 G:

(5.5.11)

276

5 Stochastic analysis by additive functionals

Proof. We first note that any exceptional set is polar by the assumption (4.2.9) and Theorem 4.1.2. As in the proof of Theorem 5.2.5, we define N Œu by (5.5.9) and then M Œu by (5.5.8). M Œu is a local AF in the strict sense and belongs to MV loc in view of Theorem 5.5.1 and Corollary 5.5.1. By thinking of v 2 F such that u D v on G, we see from (5.5.7), Theorem 5.5.2 and the optional sampling theorem that (5.5.10) and (5.5.11) hold for q.e. x 2 X and hence for any x outside a nearly Borel polar set N . On the other hand, we see from Lemma 5.5.2 (iii) and (5.5.3) that any local AF A in the strict sense satisfies A.sCt /^G D As^G C A t ^G ı s^G ;

Px -a.s. 8x 2 G:

Therefore, for any x 2 G and " > 0, Œu

Œu

Œu

Ex .M.t C"/^G  M"^G / D Ex .EX"^ G .M t ^G /I X"^G … N / D 0: Since u is bounded on G, Px -a.e. sample paths are continuous by Theorem 4.5.4 .i/ and Ex .A t ^G / is finite, i D 1; 2, we can let " # 0 in the above to obtain (5.5.10). In the same way, we have that, for x 2 G, Œu

Œu

Ex ..M.t C"/^G  M"^G /2 / D Ex .B.t C"/^G  B"^G /: By Fatou’s lemma we get Œu

Ex ..M t ^G /2 /  Ex .B t ^G / and we can proceed on the same line as in the last part of the proof of Theorem 5.2.5 to attain the identity (5.5.11). A function u 2 Floc is said to be E-harmonic on X if .u/ D ;, namely, E.u; v/ D 0;

8v 2 C

for some special standard core C of E. Corollary 5.5.2. Assume the absolute continuity condition .4:2:9/. Suppose that a function u satisfies the following conditions: (i) u 2 Fb;loc , u is E-harmonic and finely continuous on X. (ii) 1G  hui 2 S00 for any relatively compact open set G.

5.5

Then

Extensions to additive functionals locally of finite energy

Œu

D u.X t /  u.X0 /;

At

277

t < ;

is a local AF in the strict sense such that, for any relatively compact open set G, Œu

Ex .A t ^G / D 0;

8x 2 G;

Œu

Ex ..A t ^G /2 / D Ex .B t ^G /;

(5.5.12) 8x 2 G;

(5.5.13)

where B denotes a PCAF in the strict sense with Revuz measure hui . Example 5.5.1. Just as in Example 5.1.1, we consider the Brownian motion M D .X t ; Px / on Rd . Since M is a conservative diffusion, limn!1 Gn D 1 Px a.s., 8x 2 Rd , for any relatively compact open sets Gn satisfying (5.5.6) for X D Rd . Let 'i .x/ D xi ; i D 1; 2; : : : ; d , be coordinate functions. Then 1 .Rd /, D-harmonic and with co-energy measures ıij  dx. Hence ¹'i º are in Hloc Corollary 5.5.2 applies and we conclude that the processes B ti D 'i .X t /  'i .X0 /;

t > 0; 1  i  d;

are CAF’s of M in the strict sense which are local martingales with co-variation hB i ; B j i t D ıij  t Px -a.s. 8x 2 Rd : Of course, this can be seen directly using the transition function (1.4.25) and indeed ¹B ti º are continuous MAF’s in the strict sense with co-variation ıij  t : j Ex .B ti / D 0; Ex .B ti B t / D ıij  t; 8x 2 Rd . Œu 1 .Rd / looks We next examine how the CAF N t of Theorem 5.5.1 for u 2 Hloc like in the present case. If the distribution derivative u belongs to L2loc .Rd /, then Œu Nt

1 D 2

Z

t

u.Xs /ds;

.5:5:14/

0

because .u/.x/dx is then a smooth signed measure on Rd and Theorem 5.5.4 applies. In general, the following expression is valid: Z 1 t Œu 1 N t D lim u"n .Xs /ds; u 2 Hloc .Rd /; .5:5:15/ "n #0 2 0 where u" D "  u is the convolution with the mollifier " (see Exercise 1.2.1), and ¹"n º is a sequence decreasing to zero and depending only on the function u.

278

5 Stochastic analysis by additive functionals

The convergence in (5.5.15) is uniform on each finite interval of t Px -a.s. for q.e. 1 .Rd / and take, for any relatively compact open x 2 Rd . To see this, let u 2 Hloc 1 d set G, a function v 2 H .R / such that u D v on a neighbourhood of G. Then u" D v" on G for small ". Since v" converges to v in the space H 1 .Rd / as " # 0, Corollary 5.2.1 and the identity (5.5.14) for v" lead us to (5.5.15) for v, from which we get the identity (5.5.15) for u and t < G . Now (5.5.15) is clear. 1 For any u 2 Hloc .Rd /, the AF N Œu vanishes on the complement of the support of the distribution derivative u on account of Theorem 5.5.3. N Œu is of bounded variation if and only if 1=2u is represented by a signed smooth measure in the sense of (5.4.16). This condition is certainly satisfied if 1=2u is a signed Radon measure with total variation of finite energy integral. 1 For instance, the signed measure f .y/dy on R3 with f .y/ D 1B1 .y/jyjˇ sin jyj , ˇ >  52 , has the absolutely convergent energy integral based on the Newtonian 1 1 on account of (2.2.27). Consequently, the potential u.x/ D kernel v.x/ D 4 jxj R 1 3 R3 v.x  y/f .y/dy is in H .R / and, by virtue of Lemma 5.4.1, Œu Nt

Z D

t

f .Xs /ds: 0

The right side is a CAF of bounded variation of the 3-dimensional Brownian motion with a possible exceptional set being the origin 0 (see Example 5.1.1). In the case of one-dimensional Brownian motion, a non-empty exceptional set is absent and the smooth measure reduces to the positive Radon measure (Example 5.1.1). This considerably simplifies the situation. 1 Thus, in view of Theorem 5.5.4, N Œu for u 2 Hloc .R1 / is of bounded variation 00 if and only if the distribution 1=2u is a signed Radon measure or equivalently Œu u0 is of bounded variation on each finite interval. When this is the case, N t is expressed as the local time integral: Z Œu Nt D L.t; y/du0 .y/ Px -a.s. x 2 R1 : .5:5:16/ R1

In fact, from (5.1.32) we see that Z Z Z t 00 00 u" .Xs /ds D 2 L.t; y/u" .y/dy D 2 0

R1

R1

L" .t; y/du0 .y/

which combined with (5.5.15) leads us to (5.5.16). Here L" .t;  /RD "  L.t;  /. As an example, consider a function g 2 C0 .R1 / satisfying R1 g.x/dx D 0 Rx and set u.x/ D 1 g.y/dy. Then N Œu is of bounded variation if and only if the function g is of bounded variation.

5.5

Extensions to additive functionals locally of finite energy

279

In the next example we exhibit a CAF locally of zero energy but not of bounded variation. Example 5.5.2. Let M D .X t ; Px / be the 1-dimensional Brownian motion. Then the associated Dirichlet space on L2 .R1 / is . 12 D; H 1 .R1 //. Put u.x/ D x log jxj  x. Since u0 .x/ D log jxj is square integrable on any finite interval, 1 .R1 /. However u0 is not of bounded variation in the neighbouru belongs to Hloc Œu hood of the origin and hence N t is not of bounded variation in t as was noticed in the preceding example. We shall give an explicit expression of N Œu for this function. We note that the distribution derivative u00 equals Cauchy’s principal value v.p. x1 , because Z 1 1 log jxj  v 0 .x/dx E.u; v/ D 2 1  Z "  Z 1 1 0 0 log jxj  v .x/dx C log jxj  v .x/dx D lim "!0 2 1 "  Z "  Z 1 1 v.x/ v.x/ dx C dx ; D  lim "!0 2 x 1 x " for all v 2 C01 .R1 /. To give an explicit expression of N Œu , put 8 ˆ x log jxj  x  " C 12 " log "; ˆ ˆ ˆ 1=nº are nearly Borel finely open and limk!1 k D G a.s. For any x 2 Un and T < 1, 1 < n

˛ .x/

 '˛=2 .x/  Px .N G  T / C e ˛T =2 D 1  Px .N G > T / C e ˛T =2 :

Therefore we get Px .N n > T /  Px .N G > T / < 1 

1 C e ˛T =2 ; n

x 2 Un :

By taking T large enough, we find a constant 0 < ˇ < 1 such that Px .N n > T /  ˇ for q.e. x 2 Un . Let k be stopping times defined by k D inf¹t > 0I N t D kT º. Then kC1  k C 1 ı k , and consequently we have Px .N n > .k C 1/T / D Px .kC1 < n /  Ex .PX k .1 < n /I k < n /  ˇPx .k < n /  ˇ kC1 ;

5.5

Extensions to additive functionals locally of finite energy

285

for q.e. x 2 Un ; yielding that 2

Ex .N n / 

1 X

..k C 1/T /2 ˇ k < 1

q.e. x 2 Un ;

kD0

the desired property (5.5.27). Finally to show (5.5.26), we take an increasing sequence ¹G .k/ º 2 „ such that Cap.X n G .k/ / > 0. Then Px .G .k/ < 1/ > 0 q.e. by the irreducibility. Hence, .k/ for each k, there exists a sequence ¹Un º satisfying (5.5.27) for G D G .k/ . Then Sn .k/ it suffices to put Gn D kD1 Un . If M is the 1-dimensional Brownian motion, then we can represent N 2 Nc;loc by using only a single function u 2 Floc . Corollary 5.5.3. When M is the 1-dimensional Brownian motion, for any N 2 Œu Nc;loc there exists a function u 2 Floc such that N t D N t . Proof. Let ¹Gn º 2 „ and ¹un º  Floc be the sequences determined by Theorem 5.5.6. Note that every finely open set is open and every non-empty set is nonpolar for 1-dimensional Brownian motion. Hence, owing to the construction of Gn and un in the proof of Theorem 5.5.6, we may suppose that each Gn is an open interval of the form .ln ; rn / for which limn!1 ln D 1 and limn!1 rn D 1 and un .x/ D Ex .Nn /, where n D Gn . Since unC1 .x/  un .x/ D Ex .unC1 .Xn // D

rn  x x  ln unC1 .ln / C unC1 .rn /; rn  l n rn  l n

for any x 2 .ln ; rn /, the function  k  X x  ln1 rn1  x un .ln1 / C un .rn1 / u.x/ D uk .x/  rn1  ln1 rn1  ln1 nD2

is well-defined independently of k satisfying x 2 .lk ; rk / and differs from un only by a harmonic function on .ln ; rn / for each n. Hence u 2 Floc and N Œu D N . Exercise 5.5.1. Suppose that M is a conservative diffusion process and let u 2 Œu FPloc . For the decomposition of A t in (5.5.5), show the following. V then u 2 Fe and N Œu 2 Nc . (i) If M Œu 2 M, (ii) Give an example of u such that N Œu 2 Nc but u … Fe .

286

5.6

5 Stochastic analysis by additive functionals

Martingale additive functionals of finite energy and stochastic integrals

A real-valued stochastic process Z t is said to be a semi-martingale if Z t is expressible as a sum of a local martingale and a process of bounded variation. We have Œu Œu u.X0 / D M t C N t for u 2 F is not necessarily seen that the process e u.X t /  e a semi-martingales. Accordingly the transformation rule for semi-martingales due to H. Kunita and S. Watanabe (a generalized Itô’s formula) does not apply directly. Nevertheless, we showed in Lemma 5.3.3 that the energy measure of M Œu;c is identical with chui , the local part of energy measure hui , while we established in Theorem 3.2.2 a transformation rule for chui in a purely analytical manner. By translating it in terms of a new notion of stochastic integrals, we can extend a due rule of transformation for the continuous part of the martingale M Œu . We shall formulate the stochastic integral based on Theorem 5.2.1 and the Hilbertian structure V e/ of MAF’s of finite energy. of the space .M; Our assumptions on X, m, and M are the same as in 5:1. First we introduce the notion of a stochastic integral. Consider the space 

Ac D ¹A  B W A; B 2 AC c ; Ex .A t / < 1; Ex .B t / < 1; 8t > 0; q.e. x 2 Xº: 

For M , N 2 M, hM; Li defined by (5.2.9) is a unique element of Ac satisfying (5.2.10).13 Let us consider the family M1 D ¹M 2 M W hM i .2 S / is a Radon measureº: M1 is a linear space containing the family MV of MAF’s of finite energy. For M; N 2 M1 , the measure hM;Li defined by (5.2.11) is a unique signed Radon measure such that Z t  Z 1 f .x/hM;Li .dx/ D lim Em f .Xs /d hM; Lis ; 8f 2 C0 .X/: t #0 t 0 X (5.6.1) Lemma 5.6.1. If M; L 2 M1 ; f 2 L2 .XI hM i / and g 2 L2 .XI hLi /, then f  g is integrable with respect to the absolute variation jhM;Li j of hM;Li and 2

Z X 13 Cf.

jf  gjjdhM;Li j

Appendix A.3.

Z 

Z 2

X

f dhM i

X

g 2 dhLi :

(5.6.2)

5.6

287

Martingale additive functionals of finite energy and stochastic integrals

Proof. Making the expressions dhM;Li D k1 d; dhM i D k2 d and dhLi D k3 d with  D hM i C hLi C jhM;Li j, we get from (5.6.1) dhaM CbLi D .a2 k2 C 2abk1 C b 2 k3 /d

for any a; b 2 R:

Therefore, the set [

B0 D

¹x 2 X W a2 k2 .x/ C 2abk1 .x/ C b 2 k3 .x/ < 0º

a;bW rational

is -negligible and we have for each ˛; ˇ 2 R ˛ 2 f .x/2 k2 .x/ C 2˛ˇjf .x/g.x/k1 .x/j C ˇ 2 g.x/2 k3 .x/  0 for every x 2 X n B0 . Integrating this with respect to , Z Z Z 2 2 2 ˛ f dhM i C 2˛ˇ jfgj jdhM;Li j C ˇ g 2 dhLi  0; X

X

X

proving (5.6.2). Theorem 5.6.1. Given M 2 M1 and f 2 L2 .XI hM i /, there exists a unique element f  M 2 MV such that e.f  M; L/ D

1 2

Z X

f .x/hM;Li .dx/;

V 8L 2 M:

(5.6.3)

The mapping f ! f  M is linear and continuous from L2 .XI hM i / into the V e/. space .M; Proof. By virtue of Lemma 5.6.1, ˇ ˇ Z p ˇ ˇ1 1 ˇ f dhM;Li ˇˇ  p kf kL2 . hM i / e.L/; ˇ2 2 X

V L 2 M:

Hence, Theorem 5.2.1 implies Theorem 5.6.1 together with the inequality p p e.f  M /  .1= 2kf kL2 . hM i / /: The f  M 2 MV of Theorem 5.6.1 is called the stochastic integral. This terminology is legitimate by the following lemma.

288

5 Stochastic analysis by additive functionals

Lemma 5.6.2. Let M; f and f  M be as in Theorem 5:6:1. Then dhf M;Li D f  dhM;Li ;

V L 2 M:

(5.6.4)

Moreover, the following approximation holds for f 2 C0 .X/: ./

lim Ex .¹.f  M / t

jj!0

 .f  M / t º2 / D 0;

where ./

.f  M / t

D

n X

t > 0; q.e. x 2 X

(5.6.5)

f .X ti1 /.M ti  M ti1 /;

iD1

 denotes the partition 0 D t0 < t1 <    < tn D t and jj D max1i n .ti  ti1 /. Proof. Let B be a common properly exceptional set for M and hM i such that the relation (5.2.7) holds for every x 2 X n B. Applying the argument in the paragraph preceding Theorem A.3.19 to the Hunt process MjX nB , we can see for f of MjX nB satisfying for f 2 C0 .X/ that there exists a unique martingale AF M each x 2 X n B, Z t  ./ f t º2 / D 0; Ex .M f2t / D Ex f .Xs /2 d hM is : lim Ex .¹.f  M / t  M jj!0

0

(5.6.6) It follows from this that f2t / D 1 f/ D lim 1 Em .M e.M 2 t #0 2t

Z X

f 2 dhM i < 1

V f 2 M. and, consequently, M On the other hand, (5.2.10) and (5.6.6) imply Z t  f f .Xs /d hM; Lis Ex .hM ; Li t / D Ex

(5.6.7)

0

V  for any L 2 M and q.e. x 2 X. When L 2 M, e ;Li is a bounded signed hM measure and hence Z 1 f lim Ehm .hM ; Li t / D h.x/hM e ;Li .dx/ t #0 t X for any bounded -excessive function h in view of the relation (5.1.11). Therefore, we have from (5.6.7) Z Z V h.x/hM .dx/ D h.x/f .x/hM;Li .dx/; L 2 M: (5.6.8) e ;Li X

X

5.6

289

Martingale additive functionals of finite energy and stochastic integrals

h; e h 2 C0 .X/, and letting ˛ tend to infinity, we conclude that By setting h D ˛R˛e V (5.6.8) holds for any e h 2 C0 .X/. Therefore, dhM e ;Li D f  dhM;Li , 8L 2 M. R 1 f f; L/ D In particular, e.M 2 X f dhM;Li which means M D f  M . We have proved Lemma 5.6.2 when f 2 C0 .X/. On account of Lemma 5.6.1 f D f  M readily extends to f 2 and Theorem 5.6.1, the relation (5.6.8) for M 2 L .XI hM i /. Corollary 5.6.1. (i) For M 2 M1 ; f 2 L2 .XI hM i / and g 2 L2 .XI f 2 hM i /, g  .f  M / D .gf /  M:

(5.6.9)

(ii) For M; L 2 M1 , f 2 L2 .XI hM i / and g 2 L2 .XI hLi /, Z 1 f .x/g.x/hM;Li .dx/: e.f  M; g  L/ D 2 X

(5.6.10)

By using Theorem 5.1.3, we can translate the formula (5.6.4) for M 2 M1 and 

Borel f 2 L2 .XI hM i / into the following relationship for elements of Ac : V 8L 2 M:

hf  M; Li D f  hM; Li;

(5.6.11)

In this sense our stochastic integral f  M can be regarded as a variant of the stochastic integral due to M. Motoo and S. Watanabe. Note that Z t  2 Ex f .Xs / d hM is < 1 q.e. x 2 X (5.6.12) 0

for M and f as above. In fact, we have by Lemma 5.1.9 that Z

t

E 0

 f .Xs /2 d hM is

Z  .1 C t/kU1 k1 

X

f .x/2 hM i .dx/ < 1; 8 2 S00 :

Let M be in the space Mad of MAF’s in the strict sense and f be a Borel function on X satisfying (5.6.12) for every x 2 X rather than for q.e. x 2 X: Then the Motoo–Watanabe integral f .X /  M exists as a unique element in Mad satisfying condition (A.3.19) by virtue of Theorem A.3.19. Theorem A.3.20 implies that the Motoo–Watanabe integral is a strict version of the present stochastic integral f  M provided that additionally M 2 M1 and f 2 L2 .XI hM i /: Let us extend the notion of the stochastic integral f  M to a wider class of AF’s. We say that a local AF M of the process M is locally in M1 (M 2 M1;loc

290

5 Stochastic analysis by additive functionals

in notation) if there exists a sequence ¹Gn º 2 „ and a sequence ¹M .n/ º of MAF’s .n/ in M1 such that M t D M t ; t < Gn . See the latter half of the preceding section for the relevant notions and notations. The class MV loc introduced in the preceding section is a subclass of M1;loc . If M 2 M1;loc , its quadratic variation hM i 2 AC c is well-defined by hM i t D hM .n/ i t ;

t < Gn ; n D 1; 2; : : : ;

(5.6.13)

by choosing an appropriate defining set and exceptional set of hM i. hM i does not depend (up to the equivalence) on the special choice of ¹Gn º and ¹M .n/ º for M . We can still define the energy measure of M 2 M1;loc as the Revuz measure hM i of the PCAF hM i. Due to Lemma 5.1.4, we then have Z Z f .x/hM i .dx/ D f .x/hM .n/ i .dx/; f 2 Bb .X/; suppŒf   Gn : X

X

(5.6.14) In particular, Lemma 5.6.1 extends to M; L 2 M1;loc . Therefore, the formula (5.6.3) still provides us with the stochastic integral f  M 2 MV for any M 2 M1;loc and f 2 L2 .XI hM i /. The relations (5.6.9), (5.6.10) and (5.6.11) extend to M1;loc as well. In particular, .f  M / t together with hf  M i t vanishes until the first leaving time G from a relatively compact open set G, Px -a.e. for q.e. x, whenever f vanishes on the set G. Keeping this observation in mind, we can finally define the stochastic integral f  M 2 MV loc for M 2 M1;loc and f 2 L2loc .XI hM i / by the relation g  .f  M / D .gf /  M

(5.6.15)

which holds for any bounded Borel function g with compact support. In fact, it suffices to let .f  M / t D ..1Gn  f /  M / t ; t < Gn ; ¹Gn º being any sequence of relatively compact open sets such that GN n  GnC1 and Gn " X; n ! 1. Here we mention a lemma which is useful in the subsequent development. Lemma 5.6.3. Let C1 and D be a uniformly dense subfamily of C0 .X/ and an E1 -dense subfamily of F . Then the family ¹f  M Œu W f 2 C1 ; u 2 Dº of V e/. stochastic integrals is dense in .M; V RProof. Suppose that an MAF M 2 M is e-orthogonal to the above family, namely, X f dhM;M Œu i D 0; 8f 2 C1 ; 8u 2 D. This identity extends Rto all u 2 F in u.x/2 k.dx/. view of Lemma 5.6.1 and the equality hM Œu i .X/ D 2E res .u; u/C X e Hence, hM; M Œu i D 0; 8u 2 F : (5.6.16)

5.6

Martingale additive functionals of finite energy and stochastic integrals

291

In particular, (5.6.16) holds for u D R˛ g; ˛ > 0; 8g 2 C0 .X/. According to the proof of Theorem A.3.20, we then have P .M D 0/ D 1; 8 2 S00 , from which follows M D 0. Lemma 5.6.4. If the difference of u1 ; u2 2 F is constant on finely open set G, then Œu ;c Œu ;c D M t 2 ; t < G : (5.6.17) Mt 1

Proof. By Lemma 5.3.1, chu1 u2 i D 0 on G and thus hM Œu1 ;c  M Œu2 ;c i t D 0; 8t < G , which means (5.6.17). Recall the spaces Fb;loc ; Floc and FPloc introduced in 3:2 and 5:5; Fb;loc  Floc  FPloc . For u 2 FPloc , Lemma 5.6.4 guarantees to set Œu;c

Mt

Œun ;c

D Mt

;

8t < Gn ;

(5.6.18)

where ¹Gn º 2 „ and ¹un º are functions in F such that u D un on Gn . In particular, we obtain from Lemma 5.6.4 u1 ; u2 2 FPloc ; u1  u2 D constant

)

M Œu1 ;c D M Œu2 ;c :

(5.6.19)

We can now present our transformation rule in the space MV loc . Theorem 5.6.2. (i) For any ˆ 2 C 1 .Rm / and any u1 ; u2 ; : : : ; um 2 Fb;loc , the composite function ˆ.u/ D ˆ.u1 ; u2 ; : : : ; um / is again locally in Fb and M

Œˆ.u/;c

D

m X

ˆxi .e u/  M Œui ;c :

(5.6.20)

iD1

(ii) For any ˆ 2 C 1 .Rm / with bounded derivatives and any u1 ; u2 ; : : : ; um 2 Floc , the composite function ˆ.u/ is again locally in F and the formula .5:6:20/ is satisfied. (iii) Suppose ˆ.u/ 2 F in the case (i) or (ii). Then E .c/ .ˆ.u/; ˆ.u// D

m Z 1 X ˆxi .e u/ˆxj .e u/dchui ;uj i : 2 X i;j D1

(5.6.21)

292

5 Stochastic analysis by additive functionals

Proof. By virtue of Theorem 3.2.2, we have ˆ.u/ 2 Fb and Z X

fgdchˆ.u/;vi D

m Z X iD1 X

fgˆxi .e u/dchui ;vi ;

8f; g 2 C0 .X/; 8v 2 Fb

for any ˆ 2 C 1 .Rm / with ˆ.0/ D 0 and for any u1 ; u2 ; : : : ; um 2 Fb . This identity readily extends to any ˆ 2 C 1 .Rm / and any u1 ; u2 ; : : : ; um 2 Fb;loc with the help of (5.6.14) and (5.6.19). The isometry (5.6.10) and the orthogonality of the decomposition (5.3.2) give e.f  M Œˆ.u/;c ; g  M Œv / D e

m X

 f ˆxi .e u/  M Œui ;c ; g  M Œv :

iD1

On account of Lemma 5.6.3, we have the following identity of stochastic integrals V in M: f  M Œˆ.u/;c D

m X ¹f ˆxi .e u/º  M Œui ;c ;

8f 2 C0 .X/;

iD1

which in turn means (5.6.20) by virtue of (5.6.15). We have proved (i). (ii) can be proved in the same way. Under the condition of (iii), the energy of M Œˆ.u/;c equals E .c/ .ˆ.u/; ˆ.u// by Theorem 5.3.1. Thus the expression (5.6.21) results from the calculation of energy of both sides of (5.6.20). Theorem 5.6.2 immediately leads us to the following presentation of the Beurling–Deny second formula: Corollary 5.6.2. Let X be an Euclidean domain D  Rd . Suppose that the coordinate functions xi are locally in F .1  i  d /. Then (i) C 1 .D/  Floc and, for any u 2 C 1 .D/, M Œu;c D

n X

uxi  M .i /;c ;

M .i /;c D M Œxi ;c :

(5.6.22)

iD1

(ii) C01 .D/  F and, for any u 2 C01 .D/, E

.c/

.u; v/ D

d Z X i;j D1 D

uxi vxj dij ;

1 ij D chxi ;xj i : 2

Each measure ij charges no set of zero capacity.

(5.6.23)

5.6

Martingale additive functionals of finite energy and stochastic integrals

293

V We finally state a representation theorem of the space M. Theorem 5.6.3. Let X be an Euclidean domain D  Rd and M be an m-symmetric diffusion on D and .F ; E/ its Dirichlet space. Assume that E possesses C01 .D/ as its core and that E is written as E.u; v/ D

d Z X i;j D1 D

aij .x/uxi .x/vxj .x/d m.x/;

u; v 2 C01 .D/;

(5.6.24)

where A D .aij / is a bounded non-negative definite symmetric matrix. Then the space MV can be represented by stochastic integrals based on M .i / D M Œxi  ; 1  i  d: 8 ® Pd ¯ R Pd .i / W 0, !.r/ D lims"r !.s/ and !.0/ D !.0/. Let us fix T > 0: Lemma 5.7.1. For any FT -measurable set A, Pm .rT1 AI T < / D Pm .AI T < /: Proof. We first prove this for A of the type A D ¹X t1 2 A1 ; X t2 2 A2 ; : : : ; X tk 2 Ak º; 0  t1  t2     tk  T; Ai 2 B.X/.i D 1; 2; : : : ; k/. Then Lemma 4.1.2 leads us to Pm .X t1 2 A1 ; X t2 2 A2 ; : : : ; X tk 2 Ak I T < / D Pm .X0 2 X; X t1 2 A1 ; X t2 2 A2 ; : : : ; X tk 2 Ak ; XT 2 X/ D Pm .X0 2 X; XT tk 2 Ak ; XT tk1 2 Ak1 ; : : : ; XT t1 2 A1 ; XT 2 X/ D Pm .XT tk 2 Ak ; XT tk1 2 Ak1 ; : : : ; XT t1 2 A1 I T < /:

296

5 Stochastic analysis by additive functionals

It follows from the quasi-left-continuity that for each t > 0, Pm .X t  ¤ X t / D 0. Hence the right-hand side equals Pm .X t1 .rT / 2 A1 ; X t2 .rT / 2 A2 ; : : : ; X tk .rT / 2 Ak I T < /. Let H D ¹F 2 FT W Pm .F ı rT I T < / D Pm .F I T < / º. Then H is closed under monotone increasing convergence. Hence FT0 is a subset of H , and so is FT . Theorem 5.7.1. For u 2 F Œu

At

1 Œu 1 Œu Œu D M t C .MT t .rT /  MT .rT //; 2 2

0  t  T < ; Pm -a.s. (5.7.2)

Proof. Operating rT to the formula (5.2.24), we have by Lemma 5.7.1 Œu

Œu

u.XT t /  u.XT / D M t .rT / C N t .rT / Pm -a.s. on T < :

(5.7.3)

Rt Œu When u D R1 f for f 2 L2 .XI m/, we have N t D 0 .u  f /.Xs /ds, Z t Z t Œu .u  f /.Xs ı rT /ds D .u  f /.X.T s/ /ds N t .rT / D 0

Z D

0

T

Œu

T t

Œu

.u  f /.Xs /ds D NT  NT t

Œu

Œu

Pm -a.s. on T < ; (5.7.4) Œu

and thus M t .rT / D u.XT t /  u.XT /  NT C NT t . Hence Œu

Œu

MT t .rT /  MT .rT / Œu

Œu

D u.X t /  u.XT /  NT C N t Œu

D u.X t /  u.X0 / C N t

Œu

 u.X0 / C u.XT / C NT Œu

D 2u.X t /  2u.X0 /  M t ;

which implies Theorem 5.7.1. In view of Corollary 5.2.1, the relation (5.7.4) can be extended to u 2 F . Substituting T  t to (5.7.3) for t , we have Œu

Œu

u.X t / D u.XT / C MT t .rT / C NT t .rT /:

(5.7.5)

By adding (5.7.5) to (5.2.24) 1 1 1 Œu Œu Œu Œu u.X t / D .u.X0 / C u.XT // C .M t C MT t .rT // C .N t C NT t .rT //: 2 2 2 Œu

Œu

On the other hand, since by (5.7.4) NT D N t

Œu

C NT t .rT /; we have

5.7 Forward and backward martingale additive functionals

297

Theorem 5.7.2. For u 2 F 1 1 1 Œu Œu Œu u.X t / D .u.X0 / C u.XT // C .M t C MT t .rT // C NT 2 2 2 0  t  T < ; Pm -a.s. (5.7.6) Assume that M is conservative and let G t D rT1 .F t /, 0  t  T . Then Œu M t .rT / is a .Pm ; G t /-martingale in the sense that for any A D rT1 .B/ 2 Gs , B 2 Fs and 0  s  t Œu

Œu

Œu

(5.7.7)

D Em .MsŒu 1B / D Em .MsŒu .rT /I A/:

(5.7.8)

Em .M t .rT /I A/ D Em .M t .rT /1B .rT // D Em .M t 1B /

Accordingly Theorem 5.7.1 can be viewed as a representation of the AF AŒu as the sum of a forward martingale and a backward one. In the remainder of this section, we suppose that .E; F / is a regular Dirichlet form on L2 .XI m/ with the strong local property and let M D .; M; X t ; Px / be the associated diffusion process. Under the strong locality, above formulae are extended to u 2 Floc . In view of Theorem 5.3.1 and Lemma 5.3.3 the Dirichlet form E can be represented as Z 1 dchu;vi ; u; v 2 F : (5.7.9) E.u; v/ D 2 X The diffusion process M may not be conservative; we apply Lyons–Zheng decompositions to the associated reflecting diffusions on compact sets. Let K denote the class of compact sets K satisfying that m.K/ > 0, supp.1K  m/ D K and that Z 1 K dchu;vi ; u; v 2 F ; E .u; v/ D 2 K is a closable bilinear form on L2 .XI 1K m/. Notice that, for K 2 K, E K .u; u/ is assumed to vanish whenever u 2 F vanishes m-a.e. on K. For K 2 K, we denote by F K the closure of F with respect to E K C.; /1K m . By identifying L2 .XI 1K m/ with L2 .KI m/, the pair .E K ; F K / can be regarded as a regular Dirichlet space on L2 .KI m/ which possesses the strong local property on account of Exercise 3.1.1. Then .E K ; F K / is conservative because 1K 2 F K and E K .1K ; 1K / D 0 by virtue of the strong local property of E. We set Floc;ac D ¹ 2 Floc W ch i is absolutely continuous with respect to mº

298

5 Stochastic analysis by additive functionals

and denote by . / the density of ch i with respect to the measure m for 2 Floc;ac . We set further A D ¹ 2 Floc;ac \ C.X/ W lim .x/ D 1 and the set ¹x 2 XI .x/  rº x!

belongs to K for any r > 0º:



Let Br D ¹x 2 X W .x/  rº and M .r/ D ess sup¹. /.x/ W x 2 Br º for

2 Floc;ac . We have then Lemma 5.7.2. It holds that for 2 A    2r ; sup . .X t /  .X0 //  r  6m.BRCr /` p 3 M .R C r/T 0t T (5.7.10) R 1  x2 1 2 dx. where mR D 1B  m and `.x/ D p x e PmR



R

2





Proof. Let us denote the pair .E Br ; F Br / for 2 A simply by .E r ; F r /. By virtue of Theorem 7.2.2, it admits an associated diffusion process Mr D .r ; Mr ; Pxr ; X t / on Br . Since Br is assumed to be in the class K, .E r ; F r / is conservative and we may take as Mr a conservative diffusion on account of Exercise 4.5.1. Then, we have, for R; r > 0, PmR



   sup . .X t /  .X0 //  r D PmRCr sup . .X t /  .X0 //  r R

0t T

0t T



PmRCr RCr



 sup . .X t /  .X0 //  r :

0t T

(5.7.11) The first equality follows from the fact that M and MRCr have a common part on the open set ¹x 2 X W .x/ < R C rº up to an equivalence (Theorem 4.2.7) because E and E RCr have a common part on it (Theorem 4.4.2). Since the diffusion process MRCr is conservative, it follows from the equation (5.7.2) that 1 Œ  1 Œ  Œ 

.X t /  .X0 / D M t C .MT t .rT /  MT .rT //; 2 2

PmRCr -a.e. RCr

5.7 Forward and backward martingale additive functionals

299

Hence the last expression of (5.7.11) is not greater than     2 2 Π Π RCr RCr sup M t  r C PmRCr sup M t .rT /  r PmRCr 3 3 0t T 0t T   2 Π (5.7.12) C PmRCr  MT .rT /  r RCr 3     2 2 Π Π RCr RCr  2 PmRCr sup M t  r C PmRCr sup .M t /  r 3 3 0t T 0t T by Lemma 5.7.1. According to a representation theorem for continuous martingales,14 there exists a one-dimensional Brownian motion B.t / on an extended Π e; P e F e RCr / such that M Π can be represented as M t D probability space .; x Rt B. 0 . /.Xs /ds/. Therefore we see that the last expression of (5.7.12) is dominated by   Z 2 RCr e 3 Px B.t /  r d mRCr sup  3 BRCr 0t M  .RCr/T   2r D 6m.BRCr /` p : 3 M .R C r/T Now we state a general criterion for the conservation of probability. We say that .E; F / is conservative if p t 1.x/ D 1;

m-a.e. x 2 X; t > 0:

(5.7.13)

Theorem 5.7.3. If there exist 2 A and T > 0 such that for any R > 0   r D 0; (5.7.14) lim inf m.BRCr /` p r!1 M .R C r/T then .E; F / is conservative. Proof. By Lemma 5.7.2 and the assumption (5.7.14), we get for T 0 D 49 T   PmR sup . .X t /  .X0 // D 1 0t T 0

D lim PmR r!1

6 14 Cf.



sup . .X t /  .X0 //  r

0t T 0



lim inf m.BRCr /` r!1

N. Ikeda and S. Watanabe [2].

p

r M .R C r/T



 D 0;

300

5 Stochastic analysis by additive functionals

and so pT 0 1.x/ D Px .T 0 < / D Px



 sup . .X t /  .X0 // < 1 D 1;

0t T 0

for m-a.e. x; where  is the life time of the diffusion process M. By virtue of the semi-group property we have (5.7.13). Example 5.7.1. Let E be a symmetric bilinear form on L2 .Rd I dx/ defined by E.u; v/ D

d Z @u @v 1 X aij .x/ dx; d 2 @x i @xj R

u; v 2 C01 .Rd /;

(5.7.15)

i;j D1

where the coefficients aij are locally integrable Borel functions satisfying that i) aij D aj i , ii) for any compact set K, there exists a positive constant ı.K/ such P that di;j D1 aij .x/i j  ı.K/jj2 , for any x 2 K and  2 Rd . Denote by .E; F / the closure on L2 . Rd I dx/ of the symmetric form (5.7.15). Then .E; F / is conservative if d X

aij .x/i j  k.2 C jxj/2 log.2 C jxj/jj2

i;j D1

for some constant k. In fact, by employing the function .x/ D log.2 C jxj/ 2 A,   r jBRCr j ` p M .R C r/T   r  j¹x W log.2 C jxj/  R C rºj` p k.R C r/T p d k.R C r/T  r 2 2 e 2kT .RCr/ ! 0 as r ! 1;  d  e d.RCr/ r  C1 2

1 . Here jEj denotes the volume of a set E  Rd . This provided that T < 2kd sufficient condition for conservativeness is sharp in the sense that, if aij .x/ D .2 C jxj/2 .log.2 C jxj//ˇ ıij ; ˇ > 1, we see by applying Feller’s test15 to the radial part of this operator that the corresponding diffusion is not conservative.

We next state an application of Theorem 5.7.2. For 2 A put osc .Xs / D max .Xs /  min .Xs /:

0st 15 Cf.

H. P. McKean [1].

0st

0st

5.7 Forward and backward martingale additive functionals

Lemma 5.7.3. Let 2 A. Then PmR







osc .X t /  r 

0t T

16m.BRCr /`

p

301



r

(5.7.16)

M .R C r/T

Proof. For any r; R > 0,     PmR osc .X t /  r D PmRCr osc .X t /  r R 0t T

0t T

 PmRCr RCr



 osc .X t /  r :

0t T

By Theorem 5.7.2, the right-hand side is not greater than PmRCr RCr



Π

osc M t

0t T

   Π osc M t . T /  r  r C PmRCr RCr 0t T

D 2PmRCr RCr Rt is written as B. 0 . /.Xs /ds/, Z    e RCr P PmR osc .X t /  r  2 x



Π

osc M t

0t T

 r :

Π

Since M t

0t T



BRCr

osc

0t M  .RCr/T

 B.t /  r d mRCr :

The right-hand side above is estimated by the right-hand side of (5.7.16) on account of Lemma 5.7.4 below. Lemma 5.7.4. Let .B t ; P / be the law of the Brownian motion starting at the origin. Then p   P osc Bs  r  8`.r= t /: 0st

Proof. Let s0 (resp. S0 ) be random times such that Bs0 D min0st Bs (resp. BS0 D max0st Bs ). Then     P osc Bs  r D 2P osc Bs  r; s0 < S0 0st

D 2P  2P

 

0st

sup .Bs  min Bu /  r; s0 < S0

0st

0us

0st

0us



 sup .Bs  min Bu /  r :

Since Bs  min0us Bu and jBs j have the same low, the right-hand side is dominated by 4P .sup0st Bs  r/ D 8P .B t  r/ because of the reflection principle.

302

5 Stochastic analysis by additive functionals

Example 5.7.2. Let .M; g/ be a complete Riemannian manifold and V .dx/ be its volume element. The Laplace–Beltrami operator  on M is defined by u D div.grad u/; u 2 C01 .M /. The minimal diffusion process corresponding to 12  is said to be the Brownian motion on M . Since Z div.grad u/vdV .u; v/V D  M Z .grad u; grad v/g dV ; for u; v 2 C01 .M /; D M

we can regard the Brownian motion as a diffusion process corresponding to the Dirichlet form Z 1 .grad u; grad v/g dV ; E.u; v/ D 2 M F W closure of C01 .M / with respect to E1 : For a compact set K, put .x/ D inf¹d.x; y/ W y 2 Kº, where d.x; y/ is the Riemannian distance. Then 2 Floc and 2 A because of the completeness of the Riemannian manifold M . Let Kr D ¹x 2 M W .x/  rº. Noting that .r ; r /.x/  1; m-a.e., we have from Lemma 5.7.3, Z Z     Px sup .X t /  r dV .x/ D Px osc .X t /  r dV .x/ K

0t T

K

0t T

p  16  V .Kr /  `.r= T /:

(5.7.17)

For a fixed point o 2 M , let Br D ¹x 2 M I d.o; x/  rº. We then see that if lim inf r!1

1 log V .Br / < 1; r2

(5.7.18)

then .E; F / is conservative. Indeed, the equation (5.7.18) says that for any compact set K andpa sufficiently small T there exists a sequence rn ! 1 such that V .Krn /  l.rn = T / ! 0 and thus Px .sup0t T .X t / D 1/ D 0, V -a.e. on K. Hence by the completeness of M , Px .XT 2 M / D 1 V -a.e. on M . In this case the transition function p t .x;  / is absolutely continuous with respect to V and thus the Brownian motion on M is conservative in the sense that its transition function p t satisfies (5.7.19) p t 1.x/ D 1; 8x 2 X; 8t > 0: N be the restrictions Exercise 5.7.1. Let D be an arbitrary domain in Rd and C01 .D/ to DN of functions in C01 .Rd /. Let m be the Lebesgue measure on D which is

5.7 Forward and backward martingale additive functionals

303

extended to DN by setting m.@D/ D 0. We consider the regular Dirichlet form N m/ obtained by closing the space C 1 .D/ N in the Dirichlet space .E; F / on L2 .DI 0 1 1 2 . 2 D; H .D// on L .D/. F could be a proper subspace of H 1 .D/ as was seen in Example 1.6.1. Prove that .E; F / is always conservative. As the second application of Theorem 5.7.1, we consider the tightness problem for a sequence of symmetric diffusion processes. Let E n .u; v/ be a sequence of symmetric bilinear forms on L2 .Rd ; mn / defined by E n .u; v/ D

d Z 1 X n @u @v aij d mn ; d 2 @xi @xj R

for u; v 2 C01 .Rd /;

(5.7.20)

i;j D1

where mn are everywhere dense positive Radon measures on Rd . Suppose that the n coefficients aij are Borel measurable functions such that n n D aij (i) aij

(ii) for each ball Br D ¹x 2 Rd W jxj  rº there exists a constant .r/ independent of n such that 0

d X

n aij .x/i j  .r/jj2 ;

for any .x; / 2 Br  Rd :

i;j D1

For each ball Br , let E n;r .u; v/ be the symmetric form E

n;r

d Z 1 X n @u @v .u; v/ D aij d mn ; 2 @xi @xj Br

for u; v 2 C 1 .Br /:

i;j D1

We assume the closability of .E n ; C01 .Rd // and .E n;r ; C 1 .Br // on L2 .Rd I mn / and L2 .Br I mn / respectively. The corresponding closures are denoted by .E n ; F n / and .E n;r ; F n;r /. We consider the following condition. Condition 5.7.1. There exists a constant T > 0 such that for any R > 0  ³ ² r lim sup mn .BRCr /` p D 0: r!1 n .R C r/T In view of Theorem 5.7.2, Condition 5.7.1 and Exercise 4.5.1 the diffusion processes Mn D .Pxn ; X t / corresponding to .E n ; F n / by Theorem 7.2.2 can be taken to be conservative. Hence we may assume that the probability measures Pxn live on C.Œ0; 1/ ! Rd /, the space of all continuous functions from Œ0; 1/ to Rd .

304

5 Stochastic analysis by additive functionals

For a sequence of probability measures n on Rd , we define probability measures P nn on C.Œ0; 1/ ! Rd / by Z n Pxn .  /dn .x/: P n .  / D Rd

We further consider the following conditions: Condition 5.7.2. (i) supn mn .K/ < 1 for any compact set K. (ii) n is absolutely continuous with respect to mn , say n D 'n mn , and a sequence ¹'n º satisfies that supn k'n k1;K .D supn ess supx2K j'n .x/j/ < 1 for any compact set K. (iii) ¹n º is tight. Theorem 5.7.4. Under Condition 5:7:1 and Condition 5:7:2, the sequence of probability measures ¹P nn º is tight on the space C.Œ0; 1// ! Rd / equipped with the local uniform topology. Proof. For ı > 0, let n .x/ D Pxn qh;L



sup 0s;t L;jt sjh

 jX ti  Xsi j > ı :

Here X ti is the i-th component of the diffusion process X t . Note that n n c ; 'n /mn k 'n k1;BR .qh;L ; 1BR /mn C n .BR /: .qh;L

Hence, if we can show that for any L; R > 0 n ; 1BR /mn D 0; lim sup.qh;L

h!0 n

(5.7.21)

then we arrive at the theorem on account of Condition 5.7.2 (ii) and (iii). Put T 0 D 49 T and mn;r D 1Br mn . Then   n n;RCr .qh;T sup jX ti  Xsi j > ıI ƒr (5.7.22) 0 ; 1BR /mn D Pmn;R 0s;tT 0 jtsjh

C Pmn n;R



sup

0s;tT 0 jtsjh

 jX ti  Xsi j > ıI ƒcr ;

where ƒr D ¹! W sup0t T 0 .jX t j  jX0 j/ < rº and Pxn;r is a conservative diffusion process corresponding to .E n;r ; F n;r /. Since it follows from the equation (5.7.2) that 1 1 Œx  Œx  Œx  X ti  Xsi D .M t i  MsŒxi  / C .MT 0 it .rT 0 /  MT 0 is .rT 0 //; 2 2

Pmn;RCr -a.e.; n;RCr

5.7 Forward and backward martingale additive functionals

305

the first term of the right-hand side of (5.7.16) is dominated by   Pmn;RCr sup jX ti  Xsi j > ı n;RCr 0s;tT 0 jtsjh

 Pmn;RCr n;RCr



sup

0s;tT 0 jtsjh

C Pmn;RCr n;RCr D 2Pmn;RCr n;RCr





Œxi 

jM t

sup

Œxi 

0s;tT 0 jtsjh

sup

0s;tT 0 jtsjh

 MsŒxi  j > ı

jM t

Œxi 

jM t



.rT 0 /  MsŒxi  .rT 0 /j > ı



  MsŒxi  j > ı :

(5.7.23)

Let B.t/ be a one-dimensional Brownian motion on an extended probability Rt Œx  e ;P eF e n;RCr / by which M Œxi  is represented as M t i D B. 0 ai i .Xu /du/, space .; x e n;RCr -a.e. Hence we have P x ® eW !2

sup

Œxi 

0s;tT 0 jtsjh

² eW D !2

® eW  !2

jM t

sup

0s;tT 0 jtsjh

 MsŒxi  j > ı

¯

ˇ Z t  Z ˇ ˇB ai i .Xu /du  B ˇ

sup

0s;t.RCr/T 0 jtsj.RCr/h

0

jB.t/  B.s/j > ı

¯

s 0

ˇ ³ ˇ ˇ ai i .Xu /du ˇ > ı

e n;RCr P -a.e., q.e. x: x

Therefore, denoting by .W D C.Œ0; 1/ ! Rd /; P w / the standard Wiener process and setting   w j!.t /  !.s/j > ı ; .h; R C r/ D P !2W W sup 0s;t.RCr/T 0 jtsj.RCr/h

the last term of (5.7.17) is not greater than 2mn .BRCr / .h; R C r/. On the other hand, according to Lemma 5.7.2 we have   r n c Pmn;R .ƒr /  6mn .BRCr /` p : .R C r/T Hence the right-hand side of (5.7.23) is dominated by   r ; 2mn .BRCr / .h; R C r/ C 6mn .BRCr /` p .R C r/T

306

5 Stochastic analysis by additive functionals

and consequently we get for any R > 0 n lim sup.qh;T 0 ; 1BR /mn D 0

(5.7.24)

h!0 n

by virtue of Condition 5.7.1 and Condition 5.7.2 (i). Note that by the Markov property   n i i n Pmn;R jX t  Xs j > ı D .pˇn .qh;T sup 0 /; 1BR /mn ˇs;tT 0 Cˇ jtsjh

(5.7.25)

n n n n D .qh;T 0 ; 1BR0 pˇ .1BR //mn C .qh;T 0 ; 1B c 0 pˇ .1BR //mn R



n .qh;T 0 ; 1BR0 /mn

C

.1BR ; pˇn .1B c 0 //mn : R

Therefore, it follows from (5.7.24) and Lemma 5.7.2 that, for 0  ˇ  T 0 ,   n i i jX t  Xs j > ı sup lim sup sup Pmn;R h!0

ˇs;tT 0 Cˇ jtsjh

n

n n  lim sup lim sup sup¹.qh;T 0 ; 1BR0 /mn C .1BR ; pˇ .1B c 0 //mn º D 0; R0 !1

h!0

n

R

n and consequently, limh!0 supn .qh;T 0 Cˇ ; 1BR /mn D 0, for any R > 0. By repeating this argument, (5.7.21) can be established.

Chapter 6

Transformations of forms and processes

We consider a Dirichlet form E and a symmetric Hunt process M as before. This chapter deals with three basic transformations of E corresponding to the transformations of M by means of the additive functionals studied in previous sections. Let  be a positive Radon measure on X charging no set of zero capacity and A be a PCAF of M with Revuz measure . The perturbed Dirichlet form .E ; F / is a Dirichlet form on L2 .XI m/ defined by e \ L2 .XI /; F DF

E .u; u/ D E.u; u/ C .u; u/ :

E corresponds to the canonical subprocess of M with respect to A (§6.1). L FL / of E on the support Y of  is a Dirichlet form on L2 .Y I / The trace .E; defined by e e \ L2 .Y I /; FL D F

L Y ; ujY / D E.H u; H u/ E.uj e e Y Y

e denotes the quasi support of . EL corresponds to the random time change where Y of M with respect to A (§6.2). In particular, the extended Dirichlet space is unchanged under the time change when  has the full quasi support. When M is the d -dimensional Brownian motion, its multiplicative functional ˇ Z t  Z ˇ 1 t ˇˇ r ˇˇ2 r Œ L t D exp .Xs /dXs  .X /ds s 2 0 ˇ  ˇ 0  gives rise to a drift transformation of M, by which the generator 12  (resp. the R Dirichlet form 12 jruj2 dx on L2 .Rd /) is transformed into 12  C r   r (resp. R 1 1 2 2 2 d 2 d 2 jruj  dx on L .R I  dx/). Here  is a positive function in Hloc .R /. When  decays near infinity, the drift r= forces the transformed process to move back inward. §6.3 shall treat this sort of transformation in the present general setting. Especially, if  is in the domain of L2 -generator, then the transformed process of M will be shown to be recurrent even when M is non-conservative, §6.4 presents the Donsker–Varadhan type large deviation principle for the occupation time distributions of (not necessarily conservative) symmetric Markov processes. The lower bound in it will be obtained by making use of the transformation in §6.3. Finally the large deviation from the ground state will be formulated.

308

6.1

6 Transformations of forms and processes

Perturbed Dirichlet forms and killing by additive functionals

Let .X; m/ be as in (1.1.7) and MD .X t ; Px / be an m-symmetric Hunt process whose Dirichlet space .E; F / is regular. We have introduced in §1.6 the regular Dirichlet space .E g ; F / on L2 .XI m/ for a function g in K of (1.6.14) and we have seen in §4.6 (Lemma 4.6.7) that the canonical subprocess Mg of M relative to Rt C t ; C t D 0 g.Xs /ds, is a Hunt process associated the multiplicative functional e with .E g ; F /. Clearly C t is the PCAF having  D g  m as its associated smooth measure. The aim of this section is to extend this result to general positive Radon measure  charging no set of zero capacity. In this section we shall fix such a measure  on X and consider the perturbed Dirichlet form .E ; F / on L2 .XI m/ defined by e \ L2 .XI /; F DF

E .u; v/ D E.u; v/ C .u; v/ ;

(6.1.1)

u; v 2 F :

(6.1.2)

We shall see in Theorem 6.1.1 that the transition function A t f .X t // pA t f .x/ D Ex .e

is associated with .E ; F /, where A is a PCAF with Revuz measure . We then show that .E ; F / is regular again. For any ˛ > 0 and u; v 2 F , let E˛ .u; v/ D E .u; v/ C ˛.u; v/ as usual. Lemma 6.1.1. Let  be a positive Radon measure charging no set of zero capacity. Then the symmetric form .E ; F / defined by .6:1:1/ and .6:1:2/ is a Dirichlet space relative to L2 .XI m/. If .X/ < 1, then .E ; F / is regular and more specifically any standard core of .E; F / becomes a standard core of .E ; F /. Proof. To prove the closedness of the symmetric form .E ; F /, let us take a Cauchy sequence ¹un º in .E ; F /. Then it is Cauchy in .E; F / and converges e relative to E1 -metric. Since ¹un º is Cauchy in L2 .XI / as to an element u 2 F well, there exists a function v 2 L2 .XI / such that limn!1 un D v in L2 .XI /. It suffices to prove that u D v -a.e.

6.1

Perturbed Dirichlet forms and killing by additive functionals

309

To this end, let F be a closed set such that 1F   2 S0 . Then, by taking a subsequence ¹unk º of ¹un º such that unk ! v -a.e., we have from (2.2.1) and Fatou’s lemma that Z Z ju.x/  v.x/j.dx/  lim ju.x/  unk .x/j1F .x/.dx/ F

k!1 X

 C lim

k!1

q

E1 .u  unk ; u  unk / D 0:

Hence u D v -a.e. on F and hence u D v -a.e. on X owing to Theorem 2.2.4. Therefore .E ; F / is closed and a Dirichlet space because it clearly satisfies the Markovian property. Now assume that .X/ < 1. We shall prove that any function u 2 F can be approximated by a sequence of functions of F \ C0 .X/ with respect to E1 -norm. By considering uC and u separately and making truncations if necessary, we may assume that 0  u  for some constant . According to the regularity of .E; F /, there exists a sequence ¹un º  F \ C0 .X/ such that limn!1 E1 .un  u; un  . / . / u/ D 0. Put un D .0 _ un / ^ for each n. Clearly un 2 F \ C0 .X/ and it is E1 -convergent to u as n ! 1 by Theorem 1.4.2 (v). By virtue of Theorem 2.1.4, . / . / there then exists a subsequence ¹unk º such that unk ! u q.e. and in particular . / -a.e. Therefore unk ! u in L2 .XI / by Lebesgue’s dominated convergence . / theorem, proving that unk ! u relative to E1 -norm. This proof still works in proving the assertion for a standard core C of .E; F / . / by taking a sequence ¹un º from C and setting un D " .un = / for any fixed " > 0. The finiteness condition of  for the regularity will be removed after the next theorem is proved. Let A be a PCAF of M whose Revuz measure is . We denote by N a properly exceptional set for A. By virtue of Theorem A.2.11, the canonical subprocess MA D .X tA ;  A ; Px / of MjX nN with respect to the multiplicative functional .e A t / is a Hunt process on X n N . Moreover, owing to Theorem A.2.9, MA is extended to a Hunt process on X by making each point of N a trap. The transition function p A of MA is given by A t f .X t //; pA t f .x/ D Ex .e

x 2 X n N ; f 2 B C .X/:

We denote by R˛A f the Laplace transform of p A t f:  Z 1 A A t ˛t e f .X t /dt ; R˛ f .x/ D Ex 0

x 2 X n N:

.6:1:3/

.6:1:4/

310

6 Transformations of forms and processes

Lemma 4.6.7 is extended to the following Theorem 6.1.1. Let  be a positive Radon measure charging no set of zero capacity and .A t / t 0 be its associated PCAF of M. Then MA is a Hunt process associated with the Dirichlet form .E ; F / on L2 .XI m/. Proof. As in the proof of Lemma 4.6.7, Theorem 6.1.1 reduces to the assertion R˛A f 2 F ;

E˛ .R˛A f; u/ D .f; u/;

f 2 L2 .XI m/; u 2 F :

.6:1:5/

Since kR˛A f kL2 .X Im/  kR˛ f kL2 .X Im/  ˛1 kf kL2 .X Im/ , we need to prove (6.1.5) only for bounded f 2 L2 .XI m/. We first prove that the relation (6.1.5) is valid when  2 S00 . According to Lemma 5.1.5, R˛A f  R˛ f C UA˛ R˛A f D 0;

˛ > 0; f 2 B C .X/:

.6:1:6/

If  is in S00 , and if f is a bounded function in L2 .XI m/ then kR˛A f k1 < 1 and UA˛ R˛A f is a quasi continuous version of the ˛-potential U˛ .R˛A f / 2 F by Lemma 5.1.3. Since kU˛ .R˛A f /k1  kR˛A f k1 kU˛ k1 < 1 and .X/ < 1, we have that e \ L2 .XI // R˛A f D R˛ f  UA˛ R˛A f 2 F .D F and that E˛ .R˛A f; u/ D E˛ .R˛ f; u/  E˛ .UA˛ R˛A f; u/ D .f; u/  .R˛A f; u/ ;

u 2 F ;

the assertion (6.1.5). For general positive Radon measure  charging no set of zero capacity, we can take by virtue of Theorem 2.2.4 and Lemma 5.1.6 an increasing sequence ¹Fn º of closed subsets of X such that Px .limn!1 X nFn < / D 0 q.e. x 2 X, and n D 1Fn   2 S00 . Since  charges no set of zero capacity, n .B/ increases to .B/ for any B 2 B.X/. Let An D 1Fn  A. Then An is a PCAF of M with Revuz measure n . Since n 2 S00 , we have for f 2 L2 .XI m/ R˛An f 2 F n ;

E˛ n .R˛An f; u/ D .f; u/;

8u 2 F n :

.6:1:7/

Clearly jR˛An f j  R˛ jf j < 1 q.e. and hence limn!1 R˛An f .x/ D R˛A f .x/ for q.e. x 2 X. For n < m, we get from (6.1.7) E˛ n .R˛An f  R˛Am f; R˛An f  R˛Am f /  .f; R˛An f  R˛Am f /;

.6:1:8/

6.1

Perturbed Dirichlet forms and killing by additive functionals

311

which converges to zero as n; m ! 1. Therefore ¹R˛An f º is E1 -convergent in F e . On the other hand, we also get from (6.1.7) and the limit function R˛A f is in F 2 An kR˛An f kL 2 .X I /  .f; R˛ f /L2 .X Im/  n

1 2 kf kL 2 .X Im/ ˛

and by Fatou’s lemma 1 kR˛A f kL2 .X I /  p kf kL2 .X Im/ ; ˛ getting R˛A f 2 F . Finally observe the estimate j.R˛An f; u/ n  .R˛A f; u/ j  kR˛An f  R˛A f kL2 .X I n /  kukL2 .X I / C j.R˛A f; u/  n j holding for u 2 L2 .XI /. The second term of the right-hand side tends to zero as n ! 1. The first term also tends to zero because we have from (6.1.8) 2 An Am kR˛An f  R˛Am f kL 2 .X I /  .f; R˛ f  R˛ f / n

and it suffices to let first m ! 1 and then n ! 1. By letting n ! 1 in (6.1.7), we arrive at the desired equation (6.1.5). Theorem 6.1.2. Let  be a positive Radon measure charging no set of zero capacity. The Dirichlet space .E ; F / on L2 .XI m/ is then regular. Furthermore, if C is a special standard core for .E; F /, then so it is for .E ; F /. Proof. This was proved in Lemma 6.1.1 when .X/ < 1. Due to Theorem 6.1.1, we now have the canonical subprocess M of M relative to the multiplicative functional e A t as a Hunt process associated with .E ; F /. Let .M /G be the part of M on a relatively compact open set G and ..E /G ; .F /G / be its Dirichlet form on L2 .GI m/. By virtue of Theorem 4.4.5, 8 0 and f 2 L2 .XI /, Up;A f is quasi continuous and ˛ f 2 F ; Up;A

˛ E˛p .Up;A f; u/ D .f; u/ ;

u 2 F :

.6:1:12/

Proof. We can proceed analogously to the proof of Theorem 6.1.1. We first note that, if (6.1.12) were true, then ˛ f kL2 .X I /  kUp;A

1 kf kL2 .X I / : p

.6:1:13/

Hence we need to show the quasi continuity and (6.1.12) only for those functions f which are dense in L2 .XI /.

6.1

Perturbed Dirichlet forms and killing by additive functionals

313

˛ f ) is Suppose that  2 S00 and f is bounded. Then UA˛ f (resp. UA˛ Up;A bounded and a quasi continuous version of the ˛-potential of the finite measure ˛ ˛ f ). Consequently we conclude from (6.1.11) that Up;A f 2F f  (resp. Up;A and ˛ ˛ E˛ .Up;A f; u/ D E˛ .U˛ .f  /; u/  pE˛ .U˛ .Up;A f  /; u/ ˛ D .f; u/  p.Up;A f; u/ ;

u2F

the validity of (6.1.12). In general, we find as in the proof of Theorem 6.1.1 an increasing sequence ¹Fn º of quasi-closed sets such that n D 1Fn   2 S00 and n " . Let us assume that f is bounded and vanishing on X n Fn0 for some n0 . Then ˛ f 2 F n ; Up;A n

˛ E˛p n .Up;A f; u/ D .f; u/ n0 ; n

u 2 F n ;

.6:1:14/

for n  n0 , where An D 1Fn  A. Since we see from the expression (6.1.10) ˛ ˛ that Up;A f converges to Up;A f pointwise and boundedly, we can conclude from n ˛ ˛ f is (6.1.8) that Up;An f is E1 -convergent and accordingly the limit function Up;A ˛ e in F . Similarly as in the proof of Theorem 6.1.1, we can observe that Up;A f 2 ˛ ˛ L2 .XI / and that .Up;A f; u/ n converges to .Up;A f; u/ for any u 2 L2 .XI /. n Corollary 6.1.1. For any ˛  0; p  0 and f; g 2 B C , ˛ ˛ .Up;A f; g/ D .f; Up;A g/ ;

(6.1.15)

˛ g/m : .R˛pA f; g/ D .f; Up;A

(6.1.16)

The second identity follows from (6.1.5) and (6.1.12). Let us denote by Fe the extended Dirichlet space of .E ; F /. According to the regularity of .E ; F /, any function u 2 Fe possesses an .E ; F /-quasi continuous modification e u. e u is .E; F /-quasi continuous as well by Lemma 6.1.2. By virtue of Theorem 5.2.2, e u.X t /  e u.X0 / is expressed as Œu

e u.X t /  e u.X0 / D M t

V N Œu 2 Nc : M Œu 2 M;

Œu

C Nt ;

Applying the same theorem to .E ; F / and its associated canonical subprocess MA , we get the corresponding decomposition ;Œu

u.X0A / D M t e u.X tA /  e ;Œu

;Œu

C Nt

;Œu

into an MAF M t of finite energy and a CAF N t of zero energy relative to g;Œu ;Œu g;Œu ;Œu MA . We put M t D Mt and N t D Nt when  D g  m.

314

6 Transformations of forms and processes

In the proof of Theorem 5.4.1, we have already made use of the relation of N Œu and N g;Œu which is a special case of the following lemma.

Lemma 6.1.4. For any u 2 Fe , Œu Nt

D

;Œu Nt

Z C

t

0

e u.XsA /dAs ;

t <  A:

.6:1:17/ ;Œu

D Proof. We first assume that u D R1A f; f 2 L2 .XI m/ \ Bb . Then N t Rt 1 A 0 .u  f /.Xs /ds by (5.2.26). On the other hand, since UA u is a quasi continuous version of U1 .u  / and u D R1 f  UA1 u by Lemma 5.1.5, we have from Lemma 5.4.1 Z t Z t Z t Œu 1 Nt D .R1 f  f /.Xs /ds  UA u.Xs /ds C u.Xs /dAs 0

Z D

0

0

Z

t

.u  f /.Xs /ds C

0

t

u.Xs /dAs : 0

Now it suffices to note that XsA D Xs ; 8s <  A to get (6.1.17). For general u 2 Fe take a sequence un D R1A fn ; fn 2 L2 .XI m/ \ Bb which is E -convergent to u. Since ¹un º is E-convergent to u as well, the final part of the proof of Theorem 5.2.2 then implies that there exists a subsequence ¹unk º such ;Œun 

;Œu

Œun 

Œu

k u.X tA /, N t ! Nt and N t k ! N t uniformly on that unk .X tA / ! e any finite t -interval a.s. The relation (6.1.17) now follows from the same relation for un .

Exercise 6.1.1. Suppose that M satisfies the absolute continuity condition (4.2.9). Let  be a positive Radon measure on X smooth in the strict sense and A be the PCAF in the strict sense with Revuz measure  (Theorem 5.1.7). Let M be the canonical subprocess of M with respect to the multiplicative functional (in the strict sense) e A t . Show that M also satisfies (4.2.9) and Z A r˛ .x; z/r˛A .z; y/.dz/; x; y 2 X; r˛ .x; y/ D r˛ .x; y/ C X

r˛A

where r˛ and are the resolvent densities of M and M respectively in the sense of Lemma 4.2.4.

6.2

Traces of Dirichlet forms and time changes by additive functionals

Let X; m, M and .E; F / be as in the preceding section. By virtue of Theorem 2.1.7, every function in Fe has a quasi continuous modification. Throughout

6.2

Traces of Dirichlet forms and time changes by additive functionals

315

this section every function in the space Fe is considered to be quasi continuous already. As in the preceding section, let us fix an arbitrary positive Radon measure  on X charging no set of zero capacity and A be a PCAF with Revuz measure . We denote by Y the support of ; i.e., Y is the smallest closed set outside of which e of A is defined by (5.1.21). As was seen in §5.1, Y e is  vanishes. The support Y e e e a quasi-support of , Y is nearly Borel measurable and Y  Y q.e. Y n Y is -negligible but not necessarily of zero capacity. When .E; F / is transient, we can consider the following orthogonal decomposition of the Hilbert space .Fe ; E/: 8 t º:

.6:2:5/

316

6 Transformations of forms and processes

L L Show that A D . Exercise 6.2.1. Let L be the life time of M. L is a normal right continuous strong Markov In view of Theorem A.2.12, M e. We shall designate by superscript check the notions associated with process on Y L FL / and the time changed process M. L In particular, the transition this form .E; L function and the resolvent of M are given respectively by e; pL t '.x/ D Ex .'.X t // ; x 2 Y  Z 1 Z pt e '.X t /dt D Ex RLp '.x/ D Ex 0

1

e

pA t

 .6:2:6/ '.X t /dA t :

0

0 e. Since .Y nY e/ D 0 and U 0 is -symmetric by Corol' on Y Thus RLp ' D Up;A p;A lary 6.1.1, RLp and pL t are -symmetric. Furthermore lim t #0 pL t '.x/ D '.x/; 8x 2 e; 8' 2 Cb .Y /. On account of Lemma 1.4.3 pL t gives a strongly continuous semiY group TLt on L2 .Y I /. The main theorem of this section is the following.

Theorem 6.2.1. Let M D .X t ; Px / be the Hunt process associated with a regular Dirichlet space .E; F / relative to L2 .XI m/. L is -symmetric (i) The transition function .pL t / t >0 of the time changed process M L and determines a strongly continuous semigroup .T t / t >0 on L2 .Y I /. (ii) The Dirichlet space on L2 .Y I / associated with .TLt / t 0 coincides with L FL / of .6:2:4/. .E; L FL / is regular. Furthermore, if C is a special standard core of .E; F /, (iii) .E; L FL /, where CjY D ¹ujY W u 2 Cº. then CjY is a core of .E; L FL / in the L is properly associated with .E; e is of EL 1 -capacity zero. M (iv) Y n Y L L sense that pL t ' is an E1 -quasi continuous version of T t ' for any t > 0 and ' 2 L2 .Y I /. The first assertion of this theorem was just proven. We shall prove the rest in two cases separately; the transient case and the general case.

6.2.1

Transient case

We shall first suppose that the process M is transient, i.e., Z Rf .x/ D

0

1

p t f .x/dt < 1 m-a.e.; 8f 2 B C \ L1 .XI m/:

6.2

Traces of Dirichlet forms and time changes by additive functionals

Let us consider a function space defined by 8 0 is transient and gL is its reference function. Further, rewriting the relation (6.2.9) and (6.2.10), we have L that RL 0 ' D U.'  /j e Y 2 Fe and L RL 0 '; / D E.P .U.'  // ; P v/ E. D E.U.'  /; P v/ D .'; H e Y v/ D .'; / for any

2 FLe with

D vjY -a.e., v 2 Fe .

L FL /. We make use of the reguProof of Theorem 6:2:1 (iii). The regularity of .E; larity of .E ; F / proven in Theorem 6.1.2 and an approximation of the potential 0 ˛ ' by potentials Up;A ' 2 F appearing in Lemma 6.1.3. Up;A First notice the obvious relations 8 0 a Hunt process MY on Y associated with .E; L be the resolvent kernel of MY . Then Sp ' is an E-quasi continuous version of the 2 L FL / for any Borel ' 2 L .Y I / by Theorem 4.2.3. resolvent GL p ' for .E; 0 On the other hand, Up;A ' is E-quasi continuous in view of Lemma 6.1.3 and L continuous on the proof of Lemma 6.2.3, and consequently U 0 'jY is E-quasi p;A

0 '.x/ D RLp '.x/; account of Lemma 6.2.4. Therefore noting the identity Up;A 0 L e, we get that Sp ' D U ' E-q.e. x2Y on Y . In particular, if ' 2 C0C .Y / then p;A L for E-q.e. x2Y 0 '.x/ '.x/ D lim nSn '.x/ D lim nUn;A n!1 n!1 Z 1  nA t e '.X t /dA t D lim nEx n!1

D lim Ex n!1

0 1

Z

e 0

s

 '.Xs=n /ds

D Ex .'.X //; e Y L because  e Y D 0 D inf¹t > 0 I A t > 0º Px -a.s. for E-q.e. and hence E-q.e. x 2 Y . Therefore the family of functions on Y defined by L x 2 Yº ¹' 2 B C .Y / W '.x/ D Ex .'.X //; E-q.e. e Y is a monotone class containing C0C .Y /, and consequently it coincides with B C .Y /. By taking ' D 1Y ne Y we have L // E-q.e. x 2 Y: 1Y ne Y .x/ D Ex .1Y ne Y .X e Y

6.2

Traces of Dirichlet forms and time changes by additive functionals

321

e a.s. Px for E-q.e. x 2 X. Now, applying e is finely closed, X 2 Y Since Y e Y L Lemma 6.2.4 (i), we get 1Y ne Y .x/ D 0 E-q.e. x 2 Y which is equivalent to L e/ D 0. Cap.Y nY L continuous version of GL p ' for Borel It is now clear that RLp ' is an E-quasi 2 ' 2 L .Y I /. An analogous statement is valid for pL t ' in view of the proof of Theorem 4.2.3.

6.2.2

General case

We now consider the general case that .E; F / is not necessarily transient. We shall prove Theorem 6.2.1 by the method of the perturbations. First we give a general remark. Let  be an arbitrary positive Radon measure charging no set of zero capacity. We denote by .Fe ; E  / the extended Dirichlet space of the perturbed Dirichlet form .E  ; F  / on L2 .XI m/: Lemma 6.2.5. It holds that Fe \ L2 .XI / D Fe :

.6:2:13/

Proof. If u 2 Fe ; then u 2 Fe : Take an E  -approximating sequence ¹un º  F  of u: By the proof of Lemma 1.6.7, there exists g 2 K such that u 2 Feg and ¹un º is E g -convergent to u: By the 0-order version of Theorem 2.1.4, a subsequence of ¹un º converges to u q.e. Since ¹un º is L2 .XI /-Cauchy, u 2 L2 .XI /: Conversely, suppose u 2 Fe \ L2 .XI /: We may assume that u  0: By Theorem 2.1.7, we find ¹un º  F which is E-Cauchy and convergent to u q.e. Let vn D .0 _ un / ^ u .2 F /; n  1: vn then converges to u q.e. as n ! 1; and kvn k2E  kun k2E C kuk2E which is uniformly bounded in n: By the Banach-Saks theorem, the Cesàro mean sequence ¹wn º of a suitable subsequence of ¹vn º is E-Cauchy. Since 0  wn  u; ¹wn º is an E  -approximating sequence of u, yielding u 2 Fe . Returning to the setting in the beginning of §6.2, let us consider the set D D ¹x 2 X n N W Px .A1 > 0/ > 0º;

.6:2:14/

N being the properly exceptional set for the additive functional A. Obviously e; D Y

.D c / D 0:

.6:2:15/

Lemma 6.2.6. (i) There is a properly exceptional set N1 containing N such that both D n N1 and D c n N1 are M-invariant. (ii) The Dirichlet space .F C1Dc m ; E C1Dc m / on L2 .XI m/ is regular and transient.

322

6 Transformations of forms and processes

Proof. (i) In view of Lemma 5.1.11, D D ¹x 2 X W Px . < 1/ > 0º

.6:2:16/

q.e.

e. Therefore (i) is a consequence of Lemma 4.6.4. where  is the hitting time of Y (ii) By virtue of Theorem 6.1.1, q t f .x/ D Ex .e A t 

Rt 0

1D c .Xs /ds

f .X t //

.6:2:17/

is the transition function associated with the Dirichlet form .E C1Dc m ; F C1Dc m / on L2 .XI m/. By (i), we have for x 2 X n N 1 , q t f .x/ D 1D .x/Ex .e A t f  1D .X t // C 1D c .x/e t Ex .f  1D c .X t //: In particular, q t 1.x/ < 1 for some t > 0 if x 2 D n N1 and q t 1.x/  e t if x 2 D c n N1 . Therefore this Dirichlet space is transient by Lemma 1.6.5. Its regularity was shown by Theorem 6.1.2. We have already seen in Lemma 4.6.6 and Theorem 4.6.5 that the space H e Y defined by (6.2.3) constitutes a linear subspace of Fe and it is E-orthogonal to the linear subspace Fe;X ne Y defined by (6.2.1). C1

c m

D 2 . Lemma 6.2.7. (i) H e Y \ L .XI /  Fe 2 C1D c m .u; v/; 8v 2 F C1D c m . (ii) If u 2 H e e Y \L .XI /, then E .u; v/ D E

2 Proof. Fix any u 2 H e Y \ L .XI /. Then, u D H e Y w; 9w 2 Fe , and we see from (6.2.16) that u D 1D  u q.e. This implies that 1

u 2 Fe D

c m

\L2 .XI /;

E.u; v/ D E 1Dc m .u; v/;

1

8v 2 Fe D

c m

: .6:2:18/

Indeed, we get from Theorem 1.6.1 and Lemma 6.2.6 (i), E.w; w/ D E.1D  w; 1D  w/ C E.1D c  w; 1D c  w/;

w 2F:

.6:2:19/

Choose un 2 F such that ¹un º is E-Cauchy and un ! u .2 Fe / m-a.e. Then 1D  un 2 F 1Dc m ; ¹1D  un º is E 1Dc m -Cauchy, 1D  un ! 1D  u D u m-a.e. 1 c m 1 c m and further E.u; v/ D E 1Dc m .u; v/; 8v 2 Fe D : The Hence u 2 Fe D assertions then follow from (6.2.18) and Lemma 6.2.5. L FL / on Proof of Theorem 6:2:1 in the general case. To prove the closedness of .E; 2 L sequence 'n 2 FL and choose wn 2 Fe such that L .Y I /, take any E-Cauchy wn jY D 'n . By setting un D H e Y wn , we see from Lemma 6.2.7, k'n  'm kEL D kun  um kE D kun  um kE C1Dc m : 1

6.2

Traces of Dirichlet forms and time changes by additive functionals C1

323

c m

Therefore ¹un º is a Cauchy sequence in .Fe D ; E C1Dc m / which is a Hilbert space by virtue of Lemma 6.2.6 (ii). un is then E C1Dc m -convergent to C1 c m an element u 2 Fe D . C1D c m For any v 2 Fe vanishing q.e. on Y , we have from Lemma 6.2.7 that E C1Dc m .u; v/ D lim E C1Dc m .un ; v/ n!1

D lim E .un ; v/ D lim E.un ; v/ n!1

n!1

which vanishes because v is an element of Fe;X ne Y and Theorem 4.6.5 applies. 2 Applying the same theorem to H e Y u 2 He Y \ L .XI /, E C1Dc m .H e Y u; v/ D E .H e Y u; v/ D E.H e Y u; v/ D 0: C1

c m

D vanishes q.e. on Y , we then obtain that Since v D u  H e Y u 2 Fe c m C1 D .u  H e E Y u; u  H e Y u/ D 0, proving that u D H e Y u 2 He Y . If we L set ' D ujY , then ' 2 F and

k'n  'kEL D kun  ukE D kun  ukE C1Dc m ! 0; 1

which means that FL is complete with respect to EL 1 . We next show that, for any ' 2 L2 .Y I /, RLp ' 2 FL ;

ELp .RLp '; / D .'; / ;

2 FL :

.6:2:20/

As in the last paragraph of the proof of Lemma 6.2.3, we can use Lemma 6.1.3 ˛ 0 '.x/ increases to Up;A '.x/ as to get, for non-negative ' 2 L2 .Y I /, that Up;A ˛ 0 p ˛ # 0 and that ¹Up;A 'º  F are E -Cauchy as ˛ # 0. Hence Up;A ' 2 Fe . ˛ By noting that the uniform bound (6.1.13) of L2 ./-norm of Up;A ' and by letting ˛ # 0 in the equation (6.1.12), we get 0 '; v/ D .'; v/ ; E p .Up;A

8v 2 F ;



.6:2:21/

which readily extends to v 2 Fe . 0 L ' D He The strong Markov property readily yields that Up;A Y .Rp '/ and RLp ' 2 FL . If w 2 Fe , wjY D 2 L2 .Y I /, then v D H e Y w 2 Fe by Theorem 4.6.5, and we are led from (6.2.21) to 0 ELp .RLp '; / D E p .Up;A '; v/ D .'; v/ D .'; / ;

arriving at (6.2.20).

324

6 Transformations of forms and processes

L FL / is a Dirichlet space on L2 .Y I / associated with We have proved that .E; L FL / is L The regularity and the core property for .E; the time changed process M. clear from Theorem 6.1.2 and Lemma 6.2.5. Finally we shall prove that Lemma 6.2.4 holds true in general case. Indeed, L .0/ and L and Cap C1Dc m in place of Cap its proof in transient case works for Cap .0/ C1D c m

Cap.0/ , respectively. Let A be an open subset of X with Cap.0/

.A/ < 1 and C1

c m

be its 0-order equilibrium potential. is an element of the space F.0/ D which coincides with Fe \ L2 ./ \ L2 .1D c  m/ by virtue of Lemma 6.2.5. We ˇ .0/ let ' D e ˇY : Since ' D 1 -a.e. on A \ Y and ' 2 FL ;

e .0/

e .0/

L Cap.A \ Y /  EL 1 .'; '/ D E.HYQ '; HYQ '/ C .e .0/ ; e .0/ / C1D c m

 E.e .0/ ; e .0/ / C .e .0/ ; e .0/ / C1Dc m D Cap.0/

.A/:

L Next, suppose Cap.A \ Y / < 1 for an open set A  X: For the associated equilibrium potential 2 FL ; there exists v 2 Fe such that w DW HYQ v D -a.e. on Y . As in the proof for the transient case, w D 1 q.e. on A \ YQ on C1 c m because w D 0 on D c : It account of Exercise 4.6.2. Furthermore w 2 Fe D then follows from Theorem 2.1.5 that C1D c m

Cap.0/

L .A \ YQ /  E C1Dc m .w; w/ D EL 1 . ; / D Cap.A \ Y /:

We can now combine the two inequalities obtained above with Lemma 6.1.2 and Lemma 2.1.8 to get Lemma 6.2.4 in the present general case. We can then repeat exactly the same argument as in transient case to prove (iv) of Theorem 6.2.1 in general case. L FL / on L2 .Y I / defined by (6.2.4) is called the time The Dirichlet space .E; changed Dirichlet space with respect to . e D X q.e. and H D Fe by (6.2.3) Suppose  has full quasi support, then Y e Y L L and the time changed Dirichlet form .E; F / in (6.2.4) is reduced to FL D Fe \ L2 .XI /;

L E.u; u/ D E.u; u/

u 2 FL :

.6:2:22/

Since Lemma 6.2.4 holds in general, we can further conclude that the quasi notions L FL / coincide. Consequently, Theorem 2.1.7 leads us to for .E; F / and .E; FLe D Fe ;

L E.u; u/ D E.u; u/

u 2 FLe :

.6:2:23/

Here we regard the space FLe as well as Fe to consist of quasi continuous functions. Thus .Fe ; E/ is the common extended Dirichlet space for all the time changed

6.2

Traces of Dirichlet forms and time changes by additive functionals

325

Dirichlet spaces with respect to positive Radon measures charging no set of zero capacity and possessing full quasi support. If  has full quasi support, Lemma 6.2.4 holding in the general (not necessarily transient) case implies that any  2 S is again a smooth measure relative to L FL /. We prove in the next lemma that the the time changed Dirichlet space .E; L is given by changing the time of the original PCAF of M associated PCAF of M associated with . Lemma 6.2.8. Suppose that  2 S has full quasi support. Let B t be a PCAF of M L with Revuz measure  is given with Revuz measure  2 S. Then the PCAF of M L by B t D B t . Proof. Clearly BL t is a PCAF of YL . Hence it suffices to prove, for any bounded L h and f 2 B C \ L1 ./ such -integrable -excessive function (relative to M) ˛ that UB f is bounded for some ˛ > 0, the relation Z 1  .pC /t e f .XL t /d BL t D h; hf i: .6:2:24/ lim p EL h p!1

0

In fact, any -excessive function h and f 2 B C can be written as increasing limits of functions satisfying the above conditions, and consequently Theorem 5.1.3 is applicable. Put Z 1  .pC /A .pC /A t f .x/ D Ex e f .X t /dB t : RB 0

.pC /A

.pC /A

coincides with R0 defined by (6.1.4). Similarly to If B t D t , then RB (4.4.9), we have the generalized resolvent equation: .pC /A

RB

.pC /A

0 f D UB˛ f  .p C /UpC;A UB˛ f C ˛R0

UB˛ f:

Since UB˛ f ! 0 boundedly as ˛ ! 1, this equation and the relation (6.1.16) lead us to Z 1  .pC /A EL h e .pC /t f .XL t /d BL t D hh; R f i B

0

.pC /A

D lim hh; ˛R0 ˛!1

D

0 UB˛ f i D lim ˛hR˛ UpC;A h; f i ˛!1

0 hUpC;A h; f i :

Now (6.2.24) follows from 0 lim phUpC;A h; f i D Ef  .h.X0 // D h; hf i:

p!1

326

6 Transformations of forms and processes

Applying this lemma, we can extend the representation theorem of the space MV formulated in Theorem 5.6.3 to the general case that the energy measure ij D hxi ;xj i is not necessarily absolutely continuous with respect to m. Theorem 6.2.2. Let M be an m-symmetric diffusion process on a domain D  Rd . Suppose that the associated Dirichlet space .E; F / has C01 .D/ as core and E is expressed as E.u; v/ D

d Z X i;j D1 D

uxi .x/vxj .x/dij .x/;

u; v 2 C01 .D/:

Then MV D

d °X

fi  M

.i /

W

iD1

e

d X

fi  M

iD1

.i /



D

d Z X

d Z X i;j D1 D

fi .x/fj .x/dij .x/ < 1

fi .x/fj .x/dij .x/:

±

(6.2.25)

i;j D1 D

P Proof. Put  D m C di;j D1 jij j and let A t be a PCAF of M with Revuz measure . Clearly A t  t and  has full quasi support. Making a time change of M L D .XL t ; Px / on D. If relative to A t , we get a -symmetric diffusion process M M 2 M, then the time changed functional ML t D M t is a square integrable MAF L In fact, since  t  t for any t  0, the optional sampling theorem relative to M. and the submartingale inequality imply that Ex .ML t / D 0 and Ex ..ML t /2 / < 1 for q.e. x 2 D. By virtue of Lemma 6.2.9, the Revuz measures of hM i t relative to L are the same, and consequently ML is of finite energy M and of hML i t relative to M V Furthermore, since the Revuz measure 2ij L for any M 2 M. integral relative to M .i / .i / of the CAF hML .i / ; ML .j / i t of the time changed MAF ML t D M t is absolutely V there L for any M 2 M, continuous relative to , Theorem 5.5.3 is applicable to M; exist fi ; i D 1; : : : ; d such that d Z X i;j D1

d Z X dij fi .x/fj .x/ fi .x/fj .x/dij .x/ < 1; .x/d.x/ D d D D i;j D1

ML D

d X iD1

The assertion of the theorem is clear from this.

fi  ML .i / :

6.2

Traces of Dirichlet forms and time changes by additive functionals

327

The first assertion of the next theorem has been proved in Lemma 6.2.2. L is transient. Theorem 6.2.3. (i) If M is transient, then M L is recurrent. (ii) If M is recurrent, then M Proof. Suppose that M is recurrent. Since RLp '.x/ given by (6.2.6) is a EL1 L FL /, it suffices to prove that quasi continuous modification of the resolvent of .E; 1 L R0 ' D 0 or 1 -a.e. for any ' 2 LC .Y I / by (1.6.2). Let ' 2 L1C .Y I / \ Bb . For any f 2 L1C .XI m/ \ Bb , it then follows from (6.1.16) that pA

0 0 '/ D lim .f; Up;A '/ D lim .R0 f; '/ D .R0 f; '/ : .f; U0;A p!0

p!0

By the recurrence property (1.6.2) of M; m.B/ D 0 for the set B D ¹x W 0 < R0 f .x/ < 1º: Since B is finely open, B is exceptional by Lemma 4.1.4. Hence .B/ D 0 and the right end of the above identity equals 0 or 1: Therefore the set 0 '.x/ < 1º is m-negligible and .C / D 0 by the same reason C D ¹x W 0 < U0;A L follows because RL 0 ' D U 0 ' on Y e: as above. The recurrence of M 0;A

Example 6.2.1 (Time change and killing of Brownian motion). Let M be the Brownian motion on Rd . Its Dirichlet form on L2 .Rd / equals . 12 D; H 1 .Rd //, whose extended Dirichlet space He1 .Rd / is described in relation to the space G .Rd / of (1.2.14) in Example 1.5.3, Example 1.6.2 and Exercise 1.6.2. Consider positive Radon measures m and k on Rd charging no set of zero capacity such that m is of full quasi support. Then e 1e .Rd / \ L2 .Rd I m/ \ L2 .Rd I k/; F DH E.u; v/ D

1 D.u; v/ C .u; v/k ; 2

.6:2:26/ u; v 2 F ;

is the Dirichlet space on L2 .Rd I m/ associated with the Markov process on Rd obtained from the Brownian motion M by time change and killing with respect to PCAF’s with Revuz measures m and k respectively on account of (6.2.22) and Theorem 6.1.1. The stated conditions on m are met when d m D f  dx for some Borel function f strictly positive a.e. on RRd (Corollary 4.6.1). In this case, the corresponding t PCAF is given by A t D 0 f .Xs /ds. When d D 1, we can take any positive Radon measures m and k with suppŒm D R1 in view of Example 5.1.1. Then (6.2.26) reduces to Example 1.2.2.

328

6 Transformations of forms and processes

Example 6.2.2 (A trace of the Sobolev space and symmetric Cauchy process on the boundary). Consider the half space D D ¹x D .x 0 ; xd / W x 0 2 Rd 1 ; xd > 0º  Rd , d  2, and the reflecting Brownian motion M on D (see Example 4.4.2). Its Dirichlet form E on L2 .D/ is regular and identical with . 12 D; H 1 .D//, whose extended Dirichlet space He1 .D/ is described in relation to the space G .D/ of (1.2.14) in Example 1.5.3 and Example 1.6.2. e 1e .D/ the collection of functions in He1 .D/ which are E1 If we denote by H quasi continuous on D, then we see exactly in the same manner as in Exame 1 .D/ of BLD functions of potential type on D can ple 2.3.2 that the space H 0;e be described as e 1e .D/ W u D 0 q.e. on @Dº: e 10;e .D/ D ¹u 2 H H

.6:2:27/

Let H be the space of harmonic functions (in the ordinary sense) on D with finite Dirichlet integrals and put HV D H \ He1 .D/: When d  3, we get from (1.5.26), (6.2.27) and Theorem 4.3.2 that e 1e .D/ W u D H@D uº: HV D ¹u 2 H

.6:2:28/

When d D 2, we also get from (1.6.11), (6.2.27) and Exercise 4.6.4 that e 1e .D/ W u D H@D uº: H D ¹u 2 H

.6:2:29/

Let us consider the .d  1/-dimensional Lebesgue measure  on the boundary @D; .dx/ D dx 0 ı0 .dxd /. Then  charges no set of E1 -capacity zero according to Exercise 2.2.1. The topological support of  in Rd equals @D. @D is also a quasi support of  because the hitting distribution H@D .x;  / is absolutely continuous with respect to  (with the density function being the Poisson kernel) for every x 2 D and Exercise 4.6.1 applies. On account of (1.5.26) and (1.6.11), the trace (6.2.4) of . 12 D; H 1 .D// on @D relative to  now takes the expression FL D H \ L2 .@DI /; 1 L E.'; / D D .H@D '; H@D / ; 2

';

2 FL

.6:2:30/

where denotes the restriction of a function on D to @D. We shall derive an L FL / by making use of Theorem 6.2.1. explicit description of .E; Recall that the reflecting Brownian motion M is obtained from the d -dimen.d / bt D sional Brownian motion ..X t0 ; X t /; Px / by means of the reflection X

6.2

Traces of Dirichlet forms and time changes by additive functionals

329

.d /

.X t0 ; jX t j/. Denote by L.t / the local time at 0 of the one dimensional Brow.d / nian motion X t (see Example 5.1.1). Let A be the PCAF of the process M with Revuz measure . Then .6:2:31/ A t .!/ D 4L.t; !/: In fact, by making use of the equality

Z

E.x 0 ;xd / .2L .t // D

t 0

1 2 e .xd / =2s ds; p 2 s

we can see that the formula (5.1.13) holds for 4L.t /; , and the transition function of M. L D ..X 0 ; 0/; Px /x2@D of M with respect to the The time changed process M t L is a spatially AF A t is a symmetric Cauchy process on @D. More specifically, M homogeneous Hunt process on @D (see Example 4.1.1) such that   jj 0 E0 .exp.i  X t // D exp  t ;  2 Rd 1 : .6:2:32/ 2 In fact, the inverse function  t of 4L.t/ p obeys the one-sided stable law of exponent 1=2, i.e. E0 .exp.˛ t // D exp. ˛=2t/.1 Consequently, Z 1 0 0 E0 .e i Xs /P0 . t 2 ds/ E0 .exp.i  X t // D 0

Z D

1

e jj

2 s=2

0

P0 . t 2 ds/ D e jjt =2 :

L FL / is the Dirichlet space of M L on L2 .@D/ D L2 .@DI / by TheoSince .E; rem 6.2.1, we get from (6.2.32) and (1.4.26) and the identity  Z  d2 1 d .1  cos .; // d D jj d 2 jj 2 2 Rd 1 that ² ³ Z 2 1 2 L .' . C /  '.// F D ' 2 L .@D/ W d d < 1 ; Rd 1 Rd 1 jjd  Z  d2 1 L .' . C /  './/2 d d d: E.'; '/ D d jj 4 2 Rd 1 Rd 1 .6:2:33/ An analogous relation has been derived in (1.2.19) of Example 1.2.3, when D is the unit disk on R2 . 1 Cf.

N. Ikeda and S. Watanabe [2; Chap. III.4.3].

330

6 Transformations of forms and processes

Exercise 6.2.2. Let .X t ; Px / be the 1-dimensional Brownian motion and  a positive Radon measure on R1 with suppŒ D Z .D ¹: : : ; 2; 1; 0; 1; 2; : : :º/: DeL FL / the Dirichlet form corresponding to the time changed process of note by .E; X t by the PCAF associated with . Show that ´ ˇ FL D He1 .R1 /ˇZ \ L2 .ZI /; P L E.u; v/ D 14 i;j 2Z; jij jD1 .u.i/  u.j //.v.i/  v.j //; u; v 2 FL : Exercise 6.2.3. Let .X t ; Px / be the 1-dimensional Brownian motion and  a posL FL / itive Radon measure on R1 with suppŒ D RC .D Œ0; 1//. Denote by .E; the Dirichlet form corresponding to the time changed process of X t by the PCAF associated with . Let r.x; y/ D jxj C jyj  jx  yj be the function given in Exercise 4.8.1 and let r.x; L y/ be its restriction to RC  RC . L L v/ D 12 DC .u; v/; where (i) Show that F R D He1 .RC / \ L2 .RC I / and E.u; 1 0 0 DC .u; v/ D 0 u .x/v .x/dx: (ii) For any R 1bounded measurable function on Œ0; 1/ with compact support such .y/.dy/ D 0, show that the function that 0 Z 1 L R .x/ D r.x; L y/ .y/.dy/ 0

belongs to He1 .RC / and satisfies 12 DC .RL ; '/ D . ; '/ for any ' 2 FL : Let M D .X t ; Px / be the 3-dimensional Brownian motion. For a positive L1 function V , define its scattering length by Z Rt 1 .1  Ex .e  0 V .Xs /ds //dx: .V / D lim t !1 t R3 Then for a compact set K  R3 with regular boundary, .˛1K / converges to the Newtonian capacity of K as ˛ ! 1.2 Using the time change argument, we can extend this fact to a general symmetric Hunt process and smooth measures. Let X; m; M D .X t ; Px ; / and  be as in §6.1. We denote by A D A the PCAF of M with Revuz measure : Define Z Ex .e A /.dx/: .6:2:34/ ./ D X

If M is conservative, we can then rewrite ./ as Z 1 .1  Ex .e A t //m.dx/: ./ D lim t !1 t X 2 M.

Kac [1].

.6:2:35/

6.2

Traces of Dirichlet forms and time changes by additive functionals

331

Indeed, the subprocess MA D .X tA ; PxA / of M with respect to the MF e A t is an m-symmetric Hunt process with an associated regular Dirichlet form .E ; F / by Theorem 6.1.1. A t can be also regarded as a PCAF of MA with the property3 Z t  As Ex e dAs D ExA .A t /: 0

Further the Revuz measure of A with respect to MA is equal to  again.4 Hence by Theorem 5.1.3 (iii), Z t  Z t A t As A hm; 1  E .e /i D Em e dAs D Em .A t / D h; psA 1ids: 0

0

A1 / as t ! 1, we have the equation (6.2.35). Since p A t 1.x/ converges to Ex .e

Theorem 6.2.4. Assume that M is transient and .X/ < 1: It then holds that e/; .˛/ " Cap.0/ .Y

˛ " 1;

.6:2:36/

e is a quasi support of the measure  and Cap.0/ denotes the 0-order cawhere Y pacity. L FL / be L be the time changed process of M with respect to A and .E; Proof. Let M 2 2 eI / where Y is the topological support of its Dirichlet form on L .Y I / D L .Y L L e : First note that the life time  of M is A by Exercise 6.2.1. For x 2 Y Ex .e

˛A

/ D EL x .e

˛ L

/ D 1  ˛ EL x

Z

L

e 0

˛t

 dt

D 1  ˛ RL ˛ 1.x/;

e, 1 D 1 .x/. Hence the left-hand side of where 1 is the identity function on Y e Y (6.2.36) equals ˛.1; 1  ˛ RL ˛ 1/ . 2 e L The function 1 e Y is in L .Y I / by the finiteness of . Thus if 1 e Y 2 F , then L 1/ as ˛ " 1. We see ˛.1; 1  ˛ RL ˛ 1/ is non-decreasingly convergent to E.1; from Theorem 6.2.1 that L 1/ D E.H 1; H 1/; E.1; e e Y Y

He /I  e Y 1.x/ D Ex .1 e Y .X e Y < 1/; Y

e where  e Y D inf¹t > 0 W X t 2 Y º. Hence L 1/ D E.H 1; H 1/ .˛/ D ˛.1; 1  ˛ RL ˛ 1/ " E.1; e e Y Y 3 Cf. 4 P.

Blumenthal–Getoor [1; IV, (1.18)]. Fitzsimmons and R. Getoor [2].

332

6 Transformations of forms and processes

e as ˛ " 1. Since H e Y 1.x/ D Px . e Y < 1/; E.H e Y 1; H e Y 1/ equals Cap.0/ .Y / by Theorem 4.3.3. L L Assume next that 1 e Y … F . Then lim˛!1 ˛.1; 1˛ R˛ 1/ " 1 as ˛ " 1. DeL FL /: Since .FL /e \L2 .Y I / D FL , note by .FL /e the extended Dirichlet space of .E; L L 1e u  Y does not belong to the space .F /e so that the set DYL D ¹u 2 .F /e W e L e e 1 E-q.e. on Y º is empty. Therefore Cap.0/ .Y / D 1 in view of the 0-order version of Theorem 2.1.5. The proof of the theorem is complete.

6.3

Transformations by supermartingale multiplicative functionals

Let M D .; M; X t ; Px / be an m-symmetric Hunt process whose Dirichlet form .E; F / on L2 .XI m/ is regular as in the preceding section. In this section we take as the sample space  the space D.Œ0; 1/ ! X / of right continuous functions ! W Œ0; 1/ ! X possessing left limits. Here X is the one point compactification of X: X t .!/ then denotes the t -th coordinate of !. Let us denote the resolvent ¹R˛ º˛>0 of M by  Z 1 ˛t e f .X t /dt ; f 2 Bb .X/; R˛ f .x/ D Ex 0

and define the generator A by Au D ˛u  f;

u D R˛ f:

For f 2 Bb .X/ let f Mt

Z D R˛ f .X t /  R˛ f .X0 / 

0

t

.˛R˛ f  f /.Xs /ds:

f

f

Then M t is a MAF. If R˛ f .x/ D R˛ g.x/ q.e., then M t and M tg are equivalent by the uniqueness of the semimartingale R t decomposition of R˛ f .X t /  R˛ f .X0 / Rt because 0 .˛R˛ f  f /.Xs /ds and 0 .˛R˛ g  g/.Xs /ds are continuous, in parRt Rt ticular, predictable. As a result, 0 f .Xs /ds and 0 g.Xs /ds are also equivalent. Hence for  D R˛ f; f 2 Bb .X/; the MAF Z t Œ A.Xs /ds M t D .X t /  .X0 /  0

is well defined.

333

6.3 Transformations by supermartingale multiplicative functionals

Let .x/ D R˛ f .x/; f 2 Bb .X/ with .x/ > 0 q.e. For   0 and t  0, define   Z t .X t / C  A ; ; ; e L t 1¹t n1 º  Fn0 and ¹Fn0 º " X as n ! 1. Hence the exit time n from Fn0 , n D inf¹t > 0 W X t … Fn0 º, satisfies n D X nFn0 ^  and limn!1 n D  a.s. by the quasi-left-continuity of M. Lemma 6.3.1. For each t  0 and n,   Z s Z t ^n 1 A ; e L t ^n  1 D exp  .Xu /du dMsŒ : .X0 / C  0 C 0

(6.3.3)

Proof. The right-hand side of (6.3.3) is equal to  Z s  Z t ^n 1 A exp  .Xu /du .d.Xs /  A.Xs /ds/: .X0 / C  0 0 C Since   Z d ..Xs / C / exp 

0

s

 Z D exp 

A .Xu /du C

0

s



 A .Xu /du .d.Xs /  A.Xs /ds/ a:s: C

we have (6.3.3). If  > 0, the equality holds for t instead of t ^ n . By (6.3.1), ; ; ; Ex .L t /  lim inf Ex .L t ^n /  lim inf Ex .e L t ^n / D 1; n!1

;

n!1

x 2 X:

(6.3.4)

Thus we see that ¹L t º t 0 is a supermartingale with respect to .Px ; ¹F t º/ for each x 2 X. In the remainder of this section, we simply write L for L;0 .

334

6 Transformations of forms and processes 





Let p t f .x/ D Ex .L t f .X t //; t  0; x 2 X; f 2 Bb .X/. Then ¹p t º t 0  is a transition function. The Markov process M on X generated by ¹p t º t 0 is said to be the transformed process by the supermartingale MF L : Denote by  M D .; X t ; ; Px / the transformed process. Note that the transformed process has the elements ; X t ;  in common with M. This is possible because  equals D.Œ0; 1/ ! X /. M is a right process possessing the following property:5 

Ex .F 1¹t 0 q:e:; ˛ > 0; f 2 L2 .XI m/ \ BbC º: If f 2 Cb .X/ \ L2 .XI m/ is strictly positive on X, then  D R˛ f belongs to  DC . For  D R˛ f 2 DC , let M D .; X t ; ; Px / be the transformed process  by ¹L t º t 0 . Lemma 6.3.2. Let  2 DC . Then M is  2 m-symmetric. Proof. For f; g 2 bBC .X/  Z t    .X t / A  exp  .Xs /ds f .X t /g.X0 / .p t f; g/ 2 m D E 2 m .X0 / 0   Z t    A .Xs /ds .X t /f .X t / : D Em .X0 /g.X0 / exp  0  R t A Rt Since 0 A  .Xs ı r t /ds D 0  .Xs /ds, the right-hand side equals  Z t    A  Em .X t /g.X t / exp  .Xs /ds .X0 /f .X0 / D .f; p t g/ 2 m  0 by Lemma 5.7.1. We can now consider the Dirichlet form on L2 .XI  2 m/ generated by M . Let us denote it by .E  ; F  /. According to the Beurling–Deny formula, E can be represented for u; v 2 F as Z ZZ 1 c E.u; v/ D dhu;vi C .e u.x/  e u.y//.e v .x/  e v .y//J.dxdy/ 2 X X X nd (6.3.6) Z C e u.x/e v .x/k.dx/: X

5 Cf.

M. J. Sharpe [1; (62.19)].

335

6.3 Transformations by supermartingale multiplicative functionals

The next lemma shows how E changes into E  : In particular, the killing term of the latter disappear. Theorem 6.3.1. F is contained in F  and Z 1  E .u; u/ D  2 dchui 2 X ZZ C .e u.x/  e u.y//2 .x/.y/J.dx; dy/; X X nd

u2F: (6.3.7)

Proof. For n 2 N, let .E n ; F n / be the part of the Dirichlet form of .E; F / on the fine open set Fn0 : e D 0 q:e: on X n F 0 º; F n D ¹f 2 F W f n

E n .f; g/ D E.f; g/;

f; g 2 F n :

Let Mn be the part process of M on Fn0 and ¹p nt I t  0º the transition function of Mn : First Step. First we shall show that for u 2 F n \ L1 .XI m/, the limit 1  D lim .u  E .L t u.X t /I t < n /; u/ 2 m t #0 t

(6.3.8)

exists and equals the right-hand side of (6.3.7). Put 

.u  E .L t u.X t /I t < n /; u/ 2 m 

D .u  E .u.X t /I t < n /; u/ 2 m  .E ..L t  1/u.X t /I t < n /; u/ 2 m D .I/ t  .II/ t :

(6.3.9)

By Theorem 4.4.2 1 1 lim .I/ t D lim .u  p nt u; u 2 /m D E.u; u 2 /; t #0 t t #0 t and by (6.3.3), .II/ t equals  Z  Z t exp  Eu m u.X t / 0

Put

0

 Z .III/ t D Eu m u.X t /

0

t ^n

s

(6.3.10)

  A Π.Xu /du dMs I t < n : 

 Z exp 

0

s

  A Π.Xu /du dMs I t  n : 

336

6 Transformations of forms and processes

We then have    Z t ^n  Z s A 1 1 2 Π.III/ t  Ejuj m exp  2 .Xu /du d hM is (6.3.11) t2 t 0  0 1  Ejuj m .u.X t /2 I t  n /: t By noting that A= is bounded on Fn , the first factor of the right-hand side of (6.3.11) is not greater than 1 exp.ct/  Ejuj m .hM Πi t /  exp.ct /kuk1 hi .X/; t where Z hi .dx/ D chi .dx/ C 2

X

..x/  .y//2 J.dx; dy/ C .x/2 k.dx/:

The second factor of the right-hand side of (6.3.11) equals 1t .juj; p t u2  u2 /m  1 n 2 2 2 n 2 t .juj; p t u  u /m , which converges to E.juj; u /  E .juj; u / D 0 as t ! 0, because juj and u2 are elements of F n and we can appeal to Theorem 4.4.2. Hence we see that lim t !0 .1=t/(II) t is equal to  Z s    Z t ^n 1 A Œ .Xu /du dMs exp  lim Eu m u.X t / t !0 t 0  0 and also to  Z s    Z t ^n 1 A exp  Eu m .u.X t /  u.X0 // .Xu /du dMsŒ : t !0 t 0  0 (6.3.12) Œu Œu We use the decomposition u.X t /  u.X0 / D M t C N t : Since lim

 ³2 A Œ .Xu /du dMs 0  0  Z s  Z t ^n  A Œu 2 Œ  Eu m ..N t / /  Eu m exp 2 .Xu /du d hM is ; 0  0

 ² Z Œu Eu m N t

t ^n

 Z exp 

s

we get  Z s    Z t ^n 1 A Œu Œ .Xu /du dMs exp  lim Eu m N t D 0: t !0 t 0  0

6.3 Transformations by supermartingale multiplicative functionals

337

Therefore (6.3.12) equals  Z s  Z t ^n  1 A Œu Œ exp  .Xu /du d hM ; M is : lim Eu m t !0 t 0  0 Moreover, since  Rs ˇ ˇ exp  0 ˇ ˇ

 A  .Xu /du s

ˇ  1ˇ ˇ  M < 1; ˇ

s  t ^ n ;

(6.3.12) coincides with 1 lim Eu m t !0 t

Z 0

t ^n

 d hM

Œu

;M

Œ

is

Z D

X

udhu;i :

Here, hu;i is given by dhu;i .x/ D dchu;i .x/ Z C 2 .u.x/  u.y//..x/  .y//J.dx; dy/ C u.x/.x/k.dx/: X

Combining the result obtained above with (6.3.9), (6.3.10) , we see that the limit of (6.3.8) exists and Z 2 udhu;i : D E.u; u /  X

can be written as a sum of the diffusion part Z Z Z 1 1 c c dhu;u 2 i  udhu;i D  2 dchu;ui 2 X 2 X X and the jump part ZZ ¹.u.x/  u.y//.u.x/ 2 .x/  u.y/ 2 .y// X X nd

ZZ D

 2u.x/.x/.u.x/  u.y//..x/  .y//ºJ.dx; dy/

X X nd

.u.x/  u.y//2 .x/.y/J.dx; dy/:

(6.3.13) (6.3.14) (6.3.15)

Indeed, subtracting .u.x/  u.y//2 .x/.y/ from the inside of the brace ¹  º of the left-hand side, we have .u.x/  u.y//..x/  .y//u.y/.y/  .u.x/  u.y//..x/  .y//u.x/.x/

338

6 Transformations of forms and processes

and its integral with respect to the symmetric measure J equals zero. Finally, noting the killing part disappear, equals the right-hand side of (6.3.7) for u 2 F n \ L1 .XI m/. Second Step. We shall show that, for u 2 F n \ L1 .XI m/, u 2 F  and D E  .u; u/. We see from (6.3.5) that for any  2 P .X /, P -null set of  is    also P -null. Hence F t  F t ; F  F  , where F t ; F  are the minimal  (augmented) admissible filtration of M . (6.3.5) extends to F 2 F t : In addition,  Px D Px on F0 . Therefore a nearly Borel set B of M is also nearly Borel set of M , and x 2 X is regular point of B with respect to M if and only if it is so with respect to M . In particular, Fn and Fn0 are a finely closed set and its fine interior ;0 with respect to M . Let M;0 be the part process M on Fn0 and ¹p t I t  0º its transition semigroup. From (6.3.5) for F D f .X t /1¹t 0; 1B p t .x; B c / D 0; then it follows from (6.3.5) that 1B p t .x; B c / D 0   because L t > 0; t < ; Px -a.s. for every x 2 X: This means that any ¹T t ºinvariant set is ¹T t º-invariant and accordingly m-trivial by the assumption. Thus E  is irreducible. M is a  2 m-symmetric right process on X whose Dirichlet form .E  ; F  / is not only irreducible but also recurrent by Theorem 6.3.2. M is not necessarily a Borel right process but Theorem 4.7.3 can be applied to it according to the remark at the end of §4.7. In view of (6.3.5), we readily see that the absolute continuity condition (4.2.9) for the transition function of M is inherited to the same property for the transition function of M : In a similar way, we can see that a set N  X is exceptional for M if and only if so it is for M : So the statements (i), (ii) and (iii) follow from Theorem 4.7.3. Œ

We have considered the multiplicative functional L t for  belonging to the class R. But the class R is rather restrictive. In the rest of this section, we assume that .E; F / is a regular Dirichlet space with the strong local property and that the corresponding diffusion process M is conservative. We shall then extend the class R to the class FPloc . Floc / introduced in 5:5. We aim at proving that the  transformed process of M by L t never approaches in finite time to the nodal set of  2 FPloc :

6.3 Transformations by supermartingale multiplicative functionals

341

Lemma 6.3.3. It holds that for u 2 FPloc (i) hM Œu i t .r t / D hM Œu i t ; Pm -a.e., Œu

Œu

(ii) N t .r t / D N t ; Pm -a.e., where r t is the time reversal operator defined by (5.7.1). Proof. In view of (5.2.14) we get for u 2 F hM Œu i t D lim

n X

n!1

.e u.X k t /  e u.X .k1/ t //2 n

n

in L1 .Pm /:

kD1

Hence we obtain (i). We have already proved (ii) for u 2 F in (5.7.5). By the conservativeness of M these identities can be extended to u 2 FPloc . Take any  2 FPloc such that   0; m-a.e. and m.¹ > 0º/ > 0. Set X .n/ D ¹x 2 X W n1 < e .x/ < nº and define stopping times n and  by n D inf¹t > 0 W X t … X .n/ º; Œlog 

Let M t

Œlog 

Mt

Œlog 

and N t

n!1

be functionals on ¹t <  º defined by

Œlog  .n/ 

D Mt

 D lim n :

;

Œlog 

Nt

Œlog  .n/ 

D Nt

Œlog  .n/ 

;

t  n ; n  1;

(6.3.19)

Œlog  .n/ 

where  .n/ D .. n1 _ / ^ n/. Note that M t and N t are well-defined by Theorem 5.5.1 because log  .n/ 2 FPloc . The definition (6.3.16) makes sense on account of log  .n/ D log  on X .n/ and Lemma 5.5.1. We introduce a superŒ martingale multiplicative functional L t by Œ

Lt

  1 Œlog  D exp M t  hM Œlog  i t 1¹t 0 such that 0 < 1     < 1º: Œlog   CC , log  2 FPloc and L t of (6.3.17) equals exp.M t  12 hM Œlog  i t / by For FPloc the conservativeness assumption on M:

342

6 Transformations of forms and processes

CC e  by LŒ is Lemma 6.3.4. If  2 FPloc , then the transformed diffusion process M 2  m-symmetric. Œ

Proof. Let e p t f .x/ D Ex .L t f .X t //. We get for any non-negative functions f and g, .e p t f; g/ 2 m     1 Œlog  2 Œlog  D Em  .X0 / exp M t  hM i t f .X t /g.X0 / 2    1 Œlog  2 Œlog  D Em  .X0 / exp log .X t /  log .X0 /  N t  hM it 2   f .X t /g.X0 /     1 Œlog  Œlog  D Em .X0 /.X t / exp N t  hM i t f .X t /g.X0 / : 2 By Lemma 5.7.1 and Lemma 6.3.3, the last expression equals     1 Œlog  Œlog  Em .X t /.X0 / exp N t  hM i t f .X0 /g.X t / 2 D .f; e p t g/ 2 m : Lemma 6.3.5. Let A t be a PCAF of M with Revuz measure . Then, Z CC e e  2 d; 8 2 FPloc : E  2 m .A t /  t X

Proof. In view of Theorem 2.2.4 (iii), there exists a sequence ¹Fn º of increasing compact sets satisfying (2.2.16) and (2.2.17) and 1Fn   2 S00 for each n. .n/ Consider the potential U1 .1Fn / of 1Fn  and put g` D `.U1 .1Fn /  `G`C1 .U1 .1Fn ///. Take f 2 F \ C0 .X/ such that 0  f  1 and f D 1 on Fn . Then, for any v 2 F , Z .n/ .n/ fe v 1Fn d; E1 .G1 .fg` /; v/ D .g` ; f v/ ! E1 .U1 .1Fn /; f v/ D `!1

X

R .n/ v 1Fn d D E1 .U1 .1Fn /; v/. Thus G1 .fg` / converges E1 which equals X e weakly to U1 .1Fn /, and so by the Banach–Saks theorem, we can construct a se.n/ .n/ quence h` 2 F such that G1 .f h` / converges E1 -strongly to U1 .1Fn /. Then in view of Lemma 5.1.2, Corollary 5.2.1 and Lemma 5.4.1 we see that

6.3 Transformations by supermartingale multiplicative functionals

343

Rt .n/ h` /.Xs /ds converges to 0 1Fn .Xs /dAs uniformly in t on each finite interval by taking a subsequence if necessary. Since Lemma 6.3.4 implies that the measure  2 m is excessive with respect e  , we obtain to M Z t  e e 1Fn .Xs /dAs E  2 m .A t / D lim E  2 m

Rt

0 .f

n!1

0

Z

t

e2m .f  lim lim inf E n!1 `!1 0 Z .n/  2 f h` d m:  lim lim inf t n!1 `!1

.n/ h` /.Xs /ds



X

.n/

Noting that  2 f belongs to F and that G1 h` converges E1 -weakly to U1 .1Fn / by Lemma 2.2.2, we can rewrite the last expression as .n/

lim lim inf t E1 .. 2 f /; G1 h` / D lim t E1 .. 2 f /; U1 .1Fn // n!1 Z Z 2 e e D lim t  1Fn d D t  2 d:

n!1 `!1

n!1

X

X

CC Let  2 FPloc . Let ¹Gn º 2 „ and ¹n º  F be sequences such that  D n m-a.e. on Gn as before. Then log  belongs to FPloc and Z Z Z Z 2 2 e e  dhlog i D lim  n dhlog n i D lim dhn i D dhi : n!1 G n

X

n!1 G n

X

The third equality follows from LeJan’s formula (Theorem 3.2.2). Hence, by applying Lemma 6.3.5 to A t D hM Œlog  i t , we get CC for  2 FPloc :

e  2 m .hM Œlog  i t /  thi .X/; E

(6.3.21)

CC e Lemma 6.3.6. For  2 FPloc with hi .X/ < 1, the transformed diffusion M is conservative in the sense that

e  2 m . < 1/ D 0: P

(6.3.22)



Further ¹L t º t 0 is a Px -martingale for m-a.e. x 2 X: Proof. Let us define a sequence of ¹F t º-stopping times Tn and T by Tn D inf¹t W hM Œlog  i t  nº; Then

ΠL t ^Tn

6 Cf.

T D lim Tn : n!1

Œ

is a martingale by Novikov’s theorem,6 and so Ex .Ln^Tn / D 1.

N. Ikeda and S. Watanabe [2].

344

6 Transformations of forms and processes

e x .S < / D Ex .L / holds for any ¹F t º-stopping time,7 Since the identity P S e x .n ^ Tn < / D 1: we obtain P e  2 m .T < t / D 0 for On the other hand, the inequality (6.3.18) tells us that P e e any t > 0. Therefore  D 1; P  2 m -a.e. and P x -a.e. for m-a.e. x 2 X: For  such x, the second assertion holds because Ex .L t / D 1 by (6.3.5). CC For  2 FPloc with hi .X/ < 1, put  Œlog  e Œlog N D Nt C hM Œlog  i t : t

 Œlog  fŒlog D Mt  hM Œlog  i t ; M t

Then in view of Lemma 6.3.3 and Lemma 6.3.6 we can show in the same way as in the proof of Theorem 5.7.1 that  e Œlog   N e Œlog  ; e Œlog .rT / D N N t T T t

and 1 fŒlog  1 fŒlog  fŒlog  .rT // .X0 / D M C .M T t .rT /  M log e .X t /  log e t T 2 2 e  2 m -a.e. 0  t  T; P (6.3.23) CC Lemma 6.3.7. It holds that for  2 FPloc

e2m P



 e  .X t / e .X0 / 8  _ e sup  2 Thi .X/:  .X0 / e .X t / 0t T e

(6.3.24)

Proof. According to the relation (6.3.23) and the reversibility of the measure e2m, P  e  .X t / e .X0 /  _ e sup .X0 / e .X t / 0t T e   e e e D P  2 m sup j log .X t /  log .X0 /j  

e2m P



e2m P



e2m D 2P 7 Cf.

0t T

sup 0t T



 fŒlog jM j t

  2



e2m CP

  Œlog  f : j sup jM t 2 0t T

M. J. Sharpe [1; (62.20), (23.20)].

 sup 0t T

 fŒlog jM .rT /j t

  2



6.3 Transformations by supermartingale multiplicative functionals

345

 fŒlog is a continuous loBy virtue of Lemma 6.3.6 and Girsanov’s theorem,8 M t cally square integrable martingale relative to PQx for m-a.e. x with sharp bracket being identical with hM Œlog  i t : Doob’s inequality implies that the last exprese  2 m .hM fŒlog  iT / which equals 42 E e  2 m .hM Œlog  iT /: sion is dominated by 42  E  Therefore the inequality (6.3.21) leads us to the lemma.

Theorem 6.3.4. Consider  2 FPloc such that   0; m-a.e., m.¹ > 0º/ > 0 e  D .; F ; X t ; P e x / by LŒ and hi .X/ < 1. Then the transformed process M t of (6.3.17) is  2  m-symmetric, conservative in the sense of (6.3.22) and it never approaches in finite time to the set N D ¹x 2 X W e .x/ D 0 or e .x/ D 1º in the following sense: e  2 m . < 1/ D 0: (6.3.25) P e .n/ D .; F ; X t ; P e .n/ / the transformed diffusion proProof. Let us denote by M Œlog  .n/ 

cess by L t , where  .n/ D .. n1 _ / ^ n/. Then, for any 0 < ` < n,     1 e 1 e .n/ e e  .X0 /  `; n < t D P .n/ 2  .X0 /  `; n < t P 2m  m ` ` e x on the -field Fn . e x is equal to P because P On the other hand, the right-hand side is dominated by .n/

e .n/

P

2  .n/ m

 sup 0st

 e.n/  .X t / e 

.n/

_

.X0 /

 .n/ e  .X0 / e 

.n/

.X t /

n  `

  

8 log n`

2 t hi .X/

because of Lemma 6.3.7 and Lemma 3.2.3. Hence, by letting n tend to infinity we get   1 e e  .X0 /  `;  < t D 0: P 2m ` e  2 m . < t / as ` ! 1, we attain (6.3.25), Since the left-hand side tends to P .n/

.n/

pt f D e p t f  2  m-a.e. where p t denotes the which implies that limn!1 e .n/ e e . Therefore the symmetry and conservativeness of M transition function of M follow from Lemma 6.3.4 and Lemma 6.3.6. Example 6.3.1. Let M D .; M; X t ; Px / be the Brownian motion on Rd . The associated Dirichlet form E on L2 .Rd / is given by . 12 D; H 1 .Rd //. If  is a nontrivial non-negative element in the space G .Rd / defined by (1.2.14), then  satisfies the conditions in Theorem 6.3.4. Hence we can conclude from Theorem 6.3.4 8 Cf.

N. Ikeda and S. Watanabe [2].

346

6 Transformations of forms and processes

e by that the transformed process M ΠLt

 D exp

r B 

 t

1  2

ˇ  Z tˇ ˇ r ˇ2 ˇ ˇ ˇ  ˇ .Xs /ds 1¹t 0 the semigroup R1 solvent of M, i.e., p t f .x/ D Ex .f .X t // and R˛ f .x/ D Ex . 0 e ˛t f .X t /dt /. Throughout this section, we make the following assumptions: I. (Irreducibility) If a Borel set A is p t -invariant, i.e., p t .1A f /.x/ D 1A p t f .x/ m-a.e. for any f 2 L2 .XI m/ \ Bb .X/ and t > 0, then A satisfies either m.A/ D 0 or m.X n A/ D 0. Here Bb .X/ is the space of bounded Borel functions on X. II. (Strong Feller Property) R1 .Bb .X//  Cb .X/, where Cb .X/ is the space of bounded continuous functions. 9 Donsker–Varadhan

[2].

6.4

Donsker–Varadhan type large deviation principle

347

III. (Tightness) For any  > 0, there exists a compact set K such that sup R1 1K c .x/  : x2X

Here 1K c is the indicator function of the complement of K. We notice that the irreducibility I follows from the assumption II whenever the state space X is connected in view of Exercise 4.6.3. It follows from the assumption II that the resolvent kernel R˛ .x;  /; ˛ > 0; is absolutely continuous with respect to m for each x 2 X and so is the transition function p t .x;  / on account of Theorem 4.2.4. We shall take a density function r˛ .x; y/; x; y 2 X; satisfying the properties of Lemma 4.2.4 R and define, for any Borel measure  on X, R˛ ; ˛  0; by R˛ .x/ D X r˛ .x; y/.dy/, R0 .x/ D sup˛>0 R˛ .x/; x 2 X: Let us state some remarks on the assumption III. Remark 6.4.1. (i) If R1 1 2 C1 .X/, then the assumption III holds. Here C1 .X/ is the set of continuous functions vanishing at infinity. Indeed, since Z 1 Z 1   t t e 1K c .X t /dt D Ex e 1K c .X t /dt R1 1K c .x/ D Ex 0

K c

D Ex .e  K c R1 1K c .X K c //; we have sup R1 1K c .x/ D sup R1 1K c .x/  sup R1 1.x/: x2X

x2K c

x2K c

In this case, M is non-conservative because, if p t 1 D 1 for all t > 0, then R1 1 D 1: (ii) If m.X/ < 1 and kR1 k1;1 < 1, then kR1 1K c k1  kR1 k1;1 m.K c / and the assumption III is fulfilled. Here k  k1;1 is the operator norm from L1 .XI m/ to L1 .XI m/. (iii) Let M D .X t ; Px ; / be a one-dimensional diffusion process on an open interval I D .r1 ; r2 / such that Px .X  D r1 or r2 ;  < 1/ D Px . < 1/, x 2 I; and Pa .b < 1/ > 0 for any a; b 2 I: We refer the readers to Chapter 5 of K. Itô [4] for the following properties of M: It is symmetric with respect to its canonical measure m and it satisfies I trivially and II as well. The boundary point ri of I is classified into four classes: regular boundary, exit boundary, entrance boundary and natural boundary.

348

6 Transformations of forms and processes

(a) If r2 is a regular or exit boundary, then limx!r2 R1 1.x/ D 0. (b) If r2 is an entrance boundary, then limr!r2 supx2.r1 ;r2 / R1 1.r;r2 / .x/ D 0. (c) If r2 is a natural boundary, then limx!r2 R1 1.r;r2 / .x/ D 1 and thus supx2.r1 ;r2 / R1 1.r;r2 / .x/ D 1. As a result, III is satisfied if and only if no natural boundaries are present. Exercise 6.4.1. Prove that, if the transition function ¹p t º of a Hunt process satisfies p t 1 D 1 and p t .C1 .X//  C1 .X/ for t > 0, then supx2X R1 1K c .x/ D 1 for any compact set K and thus III is not fulfilled. We set D C .A/ D ¹R˛ f W ˛ > 0; f 2 L2 .XI m/ \ CbC .X/ and f 6 0º: Here CbC .X/ denotes the set of non-negative bounded continuous functions. We note that any function in D C .A/ is strictly positive. Indeed, for a given  D R˛ f 2 D C .A/, the open set ¹ > 0º is of positive capacity because it contains the set ¹f > 0º of positive m-measure. Therefore, for some > 0, the open set O D ¹ > º is of positive capacity and hence non-exceptional. By Exercise 4.7.1, Px .O < 1/ > 0 for every x 2 X and we get .x/  Ex .e ˛ O .X O //  Ex .e ˛ O / > 0;

x 2 X:

Let L be the supermartingale multiplicative functional of M introduced in §6.3:   Z t A .X t /  (6.4.1) Lt D exp  .Xs /ds  1¹t t , L t .!/ is a member of P : We then have

6.4

Donsker–Varadhan type large deviation principle

349

Theorem 6.4.1. (i) For each open set G of P lim inf t !1

1 log P .L t 2 G; t < /   inf IE ./; 2G t

 2 P:

(ii) For each closed set K of P lim sup t !1

1 log sup Px .L t 2 K; t < /   inf IE ./: 2K t x2X 

Proof. (i) Let  D R˛ f 2 D C .A/ and  2 m 2 G. Let M D .; X t ; Px / be the transformed process of M by the functional L defined by (6.4.1). Put ˇ ˇZ ² ³ Z ˇ ˇ A Ad mˇˇ <  ; S.t; / D ! 2  W ˇˇ .x/L t .!; dx/  X  X S 0 .t; / D S.t; / \ ¹! 2  W L t .!/ 2 Gº;

 > 0;

and ²

1 ‚1 D ! 2  W lim t !1 t

Z 0

t

A .Xs .!//ds D 

Z

³ Ad m ;

X

‚2 D ¹! 2  W L t .!/ converges to  2 mº: R R A 2 Observe that X j A  j d m D X jAjd m < 1;  is bounded on each compact subset of X and Cb .X/  L1 .XI  2 d m/: Therefore we can conclude by  Theorem 6.3.3 (iii) that for i D 1; 2, Px .‚i / D 1 for any x 2 X by taking the absolute continuity of the transition function of M into account. Hence, for any x 2 X and  > 0; Px .S 0 .t; // ! 1 as t ! 1: Since 

  Z t A .X0 / P .L t 2 G; t < / D exp .Xs /ds I L t 2 G .X t / 0   Z     .X0 / 0 Ad m   E I S .t; /  exp t .X t / X  Z   0 P .S .t; //  exp t Ad m   kk1 X E

350

6 Transformations of forms and processes 

and P .S 0 .t; // !

R

d as t ! 1, we obtain Z 1 Ad m  ; lim inf log P .L t 2 G; t < /  t !1 t X

and thus lim inf t !1

X

1 log P .L t 2 G; t < /   inf E.; /: t

2D C .A/  2 m2G

The right-hand side is equal to  inf E.; / D  inf IE . 2 m/ .D  inf IE .//;

2F C

2F C

 2 m2G

 2 m2G

2G

where F C is the space of non-negative functions in F . Indeed, for f 2 F C , ˛R˛ f is E1 -convergent to f as ˛ ! 1; and for a sequence ¹fn º1 nD1  C 2 2 L .XI m/\Cb .X/ which is L -convergent to f as n ! 1, R˛ fn ! R˛ f in E1 . Thus D C .A/ is E1 -dense in F C . (ii) Define Qx;t .C / D Px .L t 2 C I t < / for any set C 2 B.P /: For u 2 C D .A/ and  > 0; we get from (6.3.4)   Z t   Au u.x/ C  .Xs /ds I t <   ; Ex exp   0 uC R Au .z/.dz//Qx;t .C /. and the left-hand side dominates exp.t sup 2C X uC Hence Z 1 Au lim sup log sup Qx;t .C /  inf sup d: (6.4.2) C u2D .A/ 2C X u C  t !1 t x2X >0

Let K be a compact set of P and set ` D sup

inf

Z

2K u2D C .A/ X >0

Au d: uC

Then, for any ı > 0 and  2 K, there exist u 2 D C .A/ and  > 0 such that R Au Au X u C d  ` C ı: Since the function u C belongs to Cb .X/, there exists R Au a neighbourhood N./ of  such that X u C d  ` C 2ı for any  2 N./: S Since 2K N./ is an open covering of K, there exist 1 ; : : : ; k in K such S that K  jkD1 N.j /. Put Nj D N.j /. We then have for 1  j  k Z Au j sup d  ` C 2ı; 2Nj X u j C  j

6.4

Donsker–Varadhan type large deviation principle

Z

and so max

1j k

inf

u2D C .A/

>0

Au d  ` C 2ı: uC

sup 2Nj

351

X

Therefore, by (6.4.2) lim sup t !1

1 1 log sup Qx;t .K/  max lim sup log sup Qx;t .Nj / t 1j k t !1 t x2X x2X Z Au inf sup  max d  ` C 2ı: C 1j k u2D .A/ 2Nj X u C  >0

(6.4.3) Z

Since 

inf

u2D C .A/

>0

X

Au d D IE ./ uC

(6.4.4)

by Theorem 6.4.2 below, the proof is completed for any compact set K. For 0 <  < 1=3, let K be a compact set such that supx2X R1 1Kc .x/  . Let V .x/ D 

1Kc .x/  R1 1Kc .x/ AR1 1Kc .x/ D : R1 1Kc .x/ C  R1 1Kc .x/ C 

On account of (6.3.4) we have  Z  Z R1 1Kc C   2: exp t V .x/.dx/ dQx;t   P X

(6.4.5)

In addition, noting that ´  1 2 > V .x/  0;

1 3 ;

and V .x/ > 1, we have Z Z Z V .x/.dx/ D V .x/.dx/ C Kc

X

K

x 2 Kc ; x 2 K ;

V .x/.dx/ 

and thus  Z   Z Z Z exp t V .x/.dx/ dQx;t  exp t P

P

X

e

t

Z P

Kc



1 .Kc /  1; 3 

V .x/.dx/  t dQx;t

 t c .K / dQx;t : exp 3

352

6 Transformations of forms and processes

Let Pı D ¹ 2 P W .Kc / > ıº. We then see from (6.4.5) that   tı ı Qx;t .P /  2 exp t  : 3 For any  > 3 let J D

1 [ nD1

Then Qx;t .J /  2e t  and so lim sup t !1

3

P n1 :

(6.4.6)

n2

e t 1  e t

1 log sup Qx;t .J /  1  : t x2X

Since Jc is compact by Lemma 6.4.1 below, we have for any closed subset K of P lim sup t !1

1 log sup Qx;t .K/ t x2X

   1 1 c D lim sup log sup Qx;t .K \ J / _ lim sup log sup Qx;t .K \ J / t !1 t t !1 t x2X x2X       inf c IE ./ _ .1  /   inf IE ./ _ .1  /: 

2K

2K\J

By letting  to 1, the proof is completed. Lemma 6.4.1. The complement Jc of the set J defined by (6.4.6) is compact in P . Proof. First note that the set .Pı /c is closed. Indeed, if n 2 .Pı /c weakly converges to , .Kc /  lim inf n .Kc /  ı n!1

Kc

is closed. Since ³ 1  3 c 1 ² \ \ 3 ; P n1  2 P W .K c 1 /  D Jc D n n2 n2

because the set

nD1

nD1

Jc

the set is closed. For any  > 0, we take n0 so large that 3=n0 < . Then sup .K c / 

2Jc

which implies that the set Jc is tight.

3 < ; n0

KDK

1 n2 0

;

6.4

353

Donsker–Varadhan type large deviation principle

Let us define the I -function on P by Z I./ D  inf u2D C .A/

>0

X

Au d; uC

 2 P:

(6.4.7)

Denote by BbC .X/ the set of non-negative bounded Borel functions on X. Let us define a function on P by 

 ˛R˛ u C  d: log uC X

Z I˛ ./ D 

inf

C u2B .X/ b

(6.4.8)

>0

Lemma 6.4.2. I˛ ./ 

I./ ; ˛

 2 P:

Proof. For u D R˛ f 2 D C .A/ and  > 0, set 

Z .˛/ D  Then, d.˛/ D d˛

Z X

log X

 ˛R˛ u C  d: uC

R˛ u  ˛R˛2 u d D ˛R˛ u C 

Z X

AR˛2 u d: ˛R˛ u C 

Since ˛R2 u  R˛ u ˛R˛2 u  R˛ u  2˛ 2 ; ˛R˛ u C  ˛ R˛ u C    Z Z Z AR˛2 u AR˛2 u AR˛2 u 1 d d  d D    2 2 2 ˛2 X ˛R˛ u C  X ˛ R˛ u C  X R˛ u C ˛ 2  Hence,

1 I./: ˛2 

 ˛R˛ u C  I./ log d   ; .1/  .˛/ D uC ˛ X Z

and thus

 ˛R˛ u C  I./ d  : log u C  ˛ X

Z 

inf

u2D C .A/

>0



354

6 Transformations of forms and processes

Since for any f 2 CbC .X/, kˇRˇ f k1  kf k1 and 0  ˇRˇ f .x/ ! f .x/ as ˇ ! 1,     Z Z ˛R˛ .ˇRˇ f / C  ˇ !1 ˛R˛ f C  d ! d: (6.4.9) log log ˇRˇ f C  f C X X Define the measure ˛ by ˛ .A/ D

Z ˛R˛ .x; A/d.x/; X

A 2 B.X/:

(6.4.10)

C 2 Given v 2 BbC .X/, take a sequence ¹gn º1 nD1  Cb .X/ \ L .XI m/ such that Z jv  gn jd.˛ C / ! 0 as n ! 1: X

We then have Z Z Z j˛R˛ v  ˛R˛ gn jd  ˛R˛ .jv  gn j/d D jv  gn jd˛ ! 0 X

X

X

as n ! 1, and thus     Z Z ˛R˛ gn C  ˛R˛ v C  n!1 d ! d: log log gn C  vC X X

(6.4.11)

Hence, combining (6.4.9) and (6.4.11)     Z Z ˛R˛ u C  ˛R˛ u C  log log d D inf d; inf uC uC u2D C .A/ X u2BbC X which implies the lemma. Lemma 6.4.3. If I./ < 1, then  is absolutely continuous with respect to m. Proof. For a > 0 and A 2 B.X/, set u.x/ D a1A .x/ C 1 2 BbC .X/, where 1A is the indicator function of the set A. Then for  > 0     Z Z ˛R˛ u C  a˛R˛ .x; A/ C ˛R˛ .x; X/ C  d D d log log uC a1A .x/ C 1 C  X X  I˛ ./: Let ˛ be the measure in (6.4.10) and c˛ D ˛ .X/. By Lemma 6.4.2 and Jensen’s inequality log .a˛ .A/ C c˛ C /  .A/ log.a C 1 C / C .Ac / log.1 C /  I./=˛;

6.4

Donsker–Varadhan type large deviation principle

355

and by letting  ! 0; log .a˛ .A/ C c˛ /  .A/ log.a C 1/  I./=˛: Since log x  x  1 for x > 0, a˛ .A/ C c˛  1  .A/ log.a C 1/  I./=˛; and thus ˛ .A/  .A/ 

I./=˛ C .A/.log.a C 1/  a/ C 1  c˛ : a

Noting that log.a C 1/  a < 0, we have for any A 2 B.X/ ˛ .A/  .A/ 

I./=˛ C .log.a C 1/  a/ C 1  c˛ ; a

and .A/  ˛ .A/ D 1  c˛ C .˛ .Ac /  .Ac // 

I./=˛ C .log.a C 1/  a/ C .1  c˛ /.a C 1/ : a

Thus we see that sup j.A/  ˛ .A/j  A2B.X /

a  log.a C 1/ C I./=˛ C .1  c˛ /.a C 1/ : a

Noting that c˛ ! 1 as ˛ ! 1, we have lim sup sup j.A/  ˛ .A/j  ˛!1 A2B.X /

a  log.a C 1/ : a

Since the right-hand side converges to 0 as a ! 0 and ˛ is absolutely continuous with respect to m; we get the lemma. Theorem 6.4.2. I./ D IE ./;  2 P : Proof. I./  IE ./: Suppose that I./ D ` < 1. Lemma 6.4.2 implies that  is absolutely continuous with respect to m. Let us denote by f its density and put p f n D f ^ n. From log.1  x/  x for 1 < x < 1; we have     Z Z ˛R˛ f n C  f n  ˛R˛ f n log log 1  fdm D fdm fnC fnC X X Z f n  ˛R˛ f n f d m;  fnC X so that

Z X

f n  ˛R˛ f n f d m  I˛ .f  m/: fnC

356

6 Transformations of forms and processes

Note that f n  ˛R˛ f n f D fnC

p f  ˛R˛ f n n  ˛R˛ f n f 1¹pf nº C f 1¹pf >nº : p n C  f C

Since the absolute value p of thepfirst (resp. second) term of the right-hand side is p dominated by . f C ˛R˛ f / f 2 L1 .XI m/ (resp. f 2 L1 .XI m/), we have p Z Z p f  ˛R˛ f f n  ˛R˛ f n f d m: fdm D lim p n!1 X fnC f C X Moreover, by letting  ! 0, Z p p p I.f  m/ f . f  ˛R˛ f /d m  I˛ .f  m/  ; ˛ X p p p which implies that f 2 D.E/ and E. f ; f /  I.f  m/. 

I./  IE ./: For  2 D C .A/ define the semi-group P t by   Z t    .X t / C  A  exp  .Xs /ds f .X t / : P t f .x/ D Ex .X0 / C  0 C 



Then P t is . C /2 m-symmetric and satisfies P t 1  1 in p view of Lemma 6.3.1 and the proof of Lemma 6.3.2. Given  D f m 2 P with f 2 D.E/, let p    Z t p A  .Xs /ds f .X t / : S t f .x/ D Ex exp  0 C Then we see by the same argument as in Theorem 6.1.1 that Z p p p p 1 p A  f d m: lim . f  S t f ; f /m D E. f ; f / C t !0 t X C Since

(6.4.12)

2  s f  2 2 f / d m D . C / P t dm C X X 2  s Z f 2   . C / P t dm C X 2 s Z Z f 2 dm D f d m;  . C / C X X p p the right-hand side of (6.4.12) is non-negative, and thus we have E. f ; f /  I.f  m/. Z

 .S t

p

Z

6.4

Donsker–Varadhan type large deviation principle

357

Exercise 6.4.2. Let E be a discrete countable set and .E; F 0 / be a regular Dirichlet form on L2 .EI m/ in Example 1.2.5. Show that for a probability measure  D ¹i ºi2E on E, q 2 X m0 1 X q 0 i qij  j0 qj i C 0i ki ; where 0i D i i ; i 2 E: I./ D 2 mi i;j 2E

i2E

Since the symmetric Markov process M is allowed to be non-conservative, the proof of the lower bound (i) in Theorem 6.4.1 becomes more difficult; for the proof, it is crucial that an irreducible symmetric Markov process can be always transformed into an ergodic Markov process by the supermartingale MF L . Donsker–Varadhan [1] utilized the Feller test to verify that an absorbing onedimensional Brownian motion can be transformed into an ergodic diffusion by a drift transform. The proof of the upper bound (ii) in Theorem 6.4.1 says that without the assumption III, the upper bound holds for any compact subset of P . Let ¹fn º 2 C0 .E/ be a non-negative increasing sequence such that fn .x/ converges to f 2 CbC .E/ for each point x. Then Z Z f  ˛R˛ f fn  ˛R˛ fn d D lim d; ˛ > 0: n!1 X R˛ fn C  X R˛ f C  Set

® ¯ O D R˛ f W ˛ > 0; f 2 C C .E/; f 6 0 ; DC .A/ b

O Then we have O D ˛R˛ f  f for  D R˛ f 2 DC .A/. and define A Corollary 6.4.1 (Extended variational formula for Dirichlet forms). For f 2 F Z O Au f 2 d m: (6.4.13) E.f; f / D sup u C  C X O u2D .A/ >0

This formula has been shown to hold for any symmetric right process in Shiozawa–Takeda [1]. Let 2 be the bottom of spectrum: ² ³ Z f 2d m D 1 : (6.4.14) 2 D inf E.f; f / W f 2 F ; X

O and  > 0, Using Corollary 6.4.1, we have for any u 2 D C .A/ 2  m- ess inf x2X

O Au .x/; uC

where m-ess inf denotes the m-essential infimum.

(6.4.15)

358

6 Transformations of forms and processes

Exercise 6.4.3. Using Corollary 6.4.1, show that 2 

1 supx2X Ex ./

:

Exercise 6.4.4. Let  be a -finite measure on X with kR˛ k1 < 1 for an ˛  0. Using Exercise 4.2.2 and (6.4.15), show that  is then smooth and satisfies a Poincaré type inequality Z e u2 d  kR˛ k1  E˛ .u; u/; u 2 F : X

Setting G D K D P in Theorem 6.4.1, we have Corollary 6.4.2. For  2 P 1 1 log P .t < / D lim log sup Px .t < / t !1 t t !1 t x2X ³ ² Z u2 d m D 1 : D  inf E.u; u/ W u 2 F ; lim

(6.4.16)

X

Let us denote by kp t kp;p the operator norm of p t from Lp .XI m/ to Lp .XI m/ and put 1 p D  lim log kp t kp;p ; 1  p  1: t !1 t Note that supx2X Px .t < / D kp t k1;1 and the right-hand side of (6.4.16) is equal to 2 by the spectral theorem. We then see from Corollary 6.4.2 that 1 D 2 :

(6.4.17)

Since kp t k2;2  kp t kp;p  kp t k1;1 , 1 < p < 1 by the symmetry of p t and Riesz–Thorin interpolation theorem10 , we can obtain Theorem 6.4.3. Under the assumptions I–III, p (1  p  1) is independent of p. Exercise 6.4.5. Prove that kp t k2;2  kp t k1;1 . The probabilistic interpretation of 1 is known. 10 Cf.

E. B. Davies [2].

6.4

Donsker–Varadhan type large deviation principle

359

Exercise 6.4.6. Prove that ° ± 1 D sup   0 W sup Ex .e  / < 1 : x2X

Moreover, we can show that supx2X Ex .e 1  / D 1. Indeed, let R f .x/ D R1 R1  Ex . 0 e t f .X t // with   0. Since 0 Ex .e t I t < /dt D Ex .e /1 , kR k1;1 < 1 if and only if supx2X Ex .e  / < 1. Assume that supx2X Ex .e 1  / < 1. Then for 0 <  < 1=kR1 k1;1 , kR1  k1;1 < 1 and 2 3 R1  D R1 C R C  2 R C    :11 1 1

Hence supx2X Ex .e .1 C/ / < 1, which is contradictory to Exercise 6.4.6. We now obtain Theorem 6.4.4.12 supx2X Ex .exp.// < 1 if and only if  < 1 : Therefore, we obtain Corollary 6.4.3. Assume I–III. Then sup Ex .exp.// < 1

if and only if

2 > 1:

x2X

Exercise 6.4.7. Let M D .X t ; Px ; / be a one-dimensional diffusion process on an interval .r1 ; r2 / as is specified in Remark 6.4.1. Show that if no natural boundaries are present, then sup

Ex .exp.// < 1 if and only if

 < 2 :

x2.r1 ;r2 /

Example 6.4.1. Let M D .X t ; Px / be the Brownian motion on Rd . Let V be a non-negative continuous function on Rd with limx!1 V .x/ DR 1. Denote by t MV D .X t ; ; PxV / the subprocess of M with respect to exp. 0 V .Xs /ds/. It 2 follows from Theorem 6.1.1 that the Dirichlet form of M on L .XI m/ is expressed by ´ R E V .u; v/ D 12 D.u; v/ C Rd uv  Vdx F V D ¹u 2 H 1 .Rd / W u 2 L2 .Rd I Vdx/º: 11 Cf. 12 Cf.

T. Kato [1; III, §6]. S. Sato [1].

360

6 Transformations of forms and processes

MV satisfies the assumptions I–III. Indeed, I follows from the irreducibility of the Rt Brownian motion and the positivity of exp. 0 V .Bs /ds/: The assumption II is satisfied.13 Denote by R1V the 1-order resolvent of MV . Then since Z R1V 1.x/ D Z D

1

   Z t e t Ex exp  V .Xs /ds dt

1

   Z t t e Eo exp  V .x C Xs /ds dt;

0

0

0

0

limx!1 R1V 1.x/ D 0 by the assumption on V and the bounded convergence theorem. Hence III follows from the remark (i) on it. Noting    Z t V .Xs /ds D PxV .t < /; Ex exp  0

we see from Corollary 6.4.2 that    Z t 1 V .Xs /ds D  inf¹E V .u; u/ W u 2 F V º: lim log Ex exp  t !1 t 0 (6.4.18) We can extend (6.4.18) to any continuous function V with limx!1 V .x/ D 1 by adding a positive constant. Let M D .X t ; Px / be the Brownian motion on Rd . A positive Radon measure  on Rd is said to be in the Kato class Kd if Z .dy/ D 0; d  3; lim sup ˛#0 x2Rd jxyj 0. Since ˛n ! 0 as n ! 1, ˇ ˇZ Z ˇ ˇ n ˇ G.x; y/1¹jyj ˇ/;

6.4

Donsker–Varadhan type large deviation principle

363

where .B/ D Px .X Y 2 BI Y < 1/; B 2 B.Y /. Therefore the equation (6.4.21) holds for any x 2 Rd . We further see from Theorem 4.3.2 and Theorem 6.2.1 that the equation (6.4.21) can be rewritten as 1 lim log Px .A 1 > ˇ/ ˇ !1 ˇ ² ³ Z 1 1 d 2 e u d D 1 ; x 2 Rd : D  inf D.u; u/ W u 2 H .R /; 2 Rd (6.4.22) Since the space C01 .Rd / is D-dense in H 1 .Rd /; we can make use of Exercise 6.4.4 to conclude that the right-hand side of (6.4.22) equals ² ³ Z 1 1 d 2  inf D.u; u/ W u 2 C0 .R /; u d D 1 : 2 Rd Thus we have the next theorem. Theorem 6.4.5. For  2 Kd1 satisfying (6.4.19) 1 log Px .A 1 > ˇ/ ˇ !1 ˇ ³ ² Z 1 1 d 2 u d D 1 ; D  inf D.u; u/ W u 2 C0 .R /; 2 Rd lim

x 2 Rd :

(6.4.23)

We see from the next exercise that the infimum in (6.4.23) is the principal eigenvalue of the time changed process. Exercise 6.4.9. Let  2 Kd1 . Show that the embedding of . 12 D; He1 .Rd // into L2 .Rd I / is compact. Example 6.4.2. Let K be a compact set of Rd .d  3/ with smooth boundary and .dx/ D 1K .x/dx. We then have  Z 1 1 1K .X t /dt > ˇ lim log Px ˇ !1 ˇ 0 ² ³ Z 1 1 d 2 D  inf D.u; u/I u 2 C0 .R /; u dx D 1 : (6.4.24) 2 K In particular, for d D 3, .dx/ D 1B.0;1/ dx, B.0; 1/ D ¹x 2 R3 W jxj  1º, ³ ² Z 2 1 1 d 2 u dx D 1 D : (6.4.25) inf D.u; u/I u 2 C0 .R /; 2 8 B.0;1/

364

6 Transformations of forms and processes

Define the operator GK on L2 .K/ by Z G.x; y/f .y/dy; GK f .x/ D K

f 2 L2 .K/:

Then the right-hand side of (6.4.24) equals the reciprocal of the principal eigenvalue of GK . Exercise 6.4.10. Prove the identity (6.4.25). Exercise 6.4.11. Let E be a discrete countable set and M D .X t ; ; Pi / be the Hunt process on E associated with the regular Dirichlet form .E; F 0 / in Example 1.2.5. Suppose that M is transient. Prove that !   Z  Cap.0/ .¹iº/ 1¹i º .X t /dt > ˇ D exp  ˇ : Pi mi 0 where Cap.0/ denotes the 0-order capacity relative to .E; F 0 /: Example 6.4.3. Let M D .X t ; Px / be the Brownian motion on Rd (d  3). Let  2 Kd1 satisfying (6.4.19). Then applying Lemma 6.4.4 and Corollary 6.4.3, we see that sup Ex .exp.A (6.4.26) 1 // < 1 x2Rd

if and only if ²

1 inf D.u; u/ W u 2 C01 .Rd /; 2

³

Z 2

Rd

u d D 1 > 1:

Combining this with Theorem 2.4.1 and Corollary 2.4.1, we can see that (6.4.26) holds if ‚1 ./ < 1=4, where ‚1 ./ is defined by (2.4.6). Exercise 6.4.12. Let M D .X t ; Px / be the Brownian motion on Rd (d  3). Let K  Rd be a compact set and define mK .  / D m.K \  /, m is the Lebesgue measure. (i) Prove that ‚1 .mK / 

2 2rK , d.d 2/

where rK is the radius of the ball with the p same volume as K, that is, rK D .m.K/. d2 C 1//1=d = .

(ii) Prove that if m.K/ < . 18 d.d  2/ /d=2 =. d2 C 1/, then ‚1 .mK / < 1=4. Exercise 6.4.13. Let M D .X t ; Px / be the Brownian motion on Rd (d  3). Let ır be the surface measure of the sphere Sr D ¹jxj D rº and `r .t / the PCAF corresponding to ır .

6.4

Donsker–Varadhan type large deviation principle

(i) Show that supx2Rd Ex .exp.`r .1/// < 1 if and only if r
0 then satisfies the large deviaThe family of probability measures ¹Q tion principle with the rate function J./ WD IE ./  2 ;  2 P , as t ! 1 in Varadhan’s formulation, where 2 is the bottom of the spectrum of the L2 e x;t º t >0 obeys the full generator A for E defined by (6.4.14). In other words, ¹Q large deviation principle with the good rate function J./. In addition, we shall see that the ground state 0 of the operator A exists and 02  m is a unique probability measure for which J./ D 0. On account of these facts, we shall reinterpret Theorem 6.4.1 as a large deviation principle from the ground state. A function 0 on X is called a ground state of the L2 -generator A for E if 0 2 F ; k0 k2 D 1 and E.0 ; 0 / D 2 . Lemma 6.4.5. Assume that M satisfies I–III. Then there exists a ground state 0 of A uniquely up to a sign. 0 can be taken to be strictly positive on X: Proof. In our proof of the existence of the minimizer in the right-hand side of (6.4.14), the identification of the I-function with the Dirichlet form (Theorem 6.4.2) plays a crucial role. In fact, let ¹un º1 nD1  F be a minimizing sequence, that is, kun k2 D 1 and 0 D limn!1 E.un ; un /. In a usual proof of the existence of the minimizer, the E1 -weak compactness of ¹un º1 nD1 in F and the lower semi-continuity of E are used, while we use the tightness of ¹u2nmº1 nD1  P and the lower semi-continuity of the function I with respect to the weak topology. We see from Theorem 2.4.2 that for any  > 0 there exists a compact set K such that Z   u2n  d m  4kR1 IK c k1  sup E.un ; un / C 1 < ; sup n

Kc

n

that is, the subset ¹u2n mº of P is tight. Hence there exists a subsequence ¹u2nk mº such that u2nk m converges weakly to a probability measure . It follows from

366

6 Transformations of forms and processes

Theorem 6.4.2 that the function IE is lower semi-continuous with respect to the weak topology, IE ./  lim inf IE .u2nk  m/ D lim inf E.unk ; unk / < 1: k!1

k!1

Therefore we see that  is expressed as  D 02  m; 0 2 F ; 0  0. 0 is just a ground state of A: It follows from the inequality k0 C gk2E  2 k0 C gk22 holding for any g 2 F and any  > 0 that E.0 ; g/ D 2 .0 ; g/: Hence ˛R˛2 0 D 0 ; ˛ > 2 ; which implies that 0 is strictly positive in view of the paragraph above (6.4.1). To prove the uniqueness of the ground state, we introduce a closed symmetric form .E 0 ; F 0 / on L2 .XI 02 m/ by ´ E 0 .u; v/ D E.u0 ; v0 /  2 .u0 ; v0 / (6.4.28) F 0 D ¹u 2 L2 .XI 02  m/ W u0 2 F º: 



Since 1 2 F 0 ; E 0 .1; 1/ D 0 and the associated resolvent G˛ 0 satisfies G˛ 0 f D 01 R˛2 .f 0 /; ˛ > 2 ; we see that .E 0 ; F 0 / is an irreducible recurrent Dirichlet form so that 15 f is constant whenever f 2 F 0 ; E 0 .f; f / D 0: Let 0 be another ground state of A: Then 0 D f 0 with f D 0 =0 2 F 0 ; E 0 .f; f / D E. 0 ; 0 /  2 D 0; which yields that f is constant and 0 D ˙0 : .X t / Exercise 6.4.14. Define the MF L t by L t D e 2 t 00.X . Show that the trans0/ formed process of M by L t generates the Dirichlet form (6.4.28).

Exercise 6.4.15. Let .X t ; Px / be the Brownian motion on Rd and D  Rd a domain with regular boundary. If D is unbounded, assume that limx2D;jxj!1 Px .D > 0/ D 0. (i) Show that the Dirichlet form .1=2D; H01 .D// admits the ground state. (ii) Denote by 0 the ground state in (i). Show that for a set B R D of finite Lebesgue measure, lim t !1 e 2 t Px .X t 2 B; t < D / D 0 .x/ B 0 .x/dx. Exercise 6.4.16. Let M D .X t ; Px / be the Brownian motion on Rd , d  3. Suppose a finite measure  2 Kd satisfies (6.4.19) and the inequality (2.4.20) for L be the time changed process of M by the ˛ D 2 and for some q > 2. Let M L Show that PCAF A with Revuz measure  andR 0 be the ground state for M:  ˇ 2 limˇ !1 e Px .A1 > ˇ/ D 0 .x/ Y 0 d; 8x 2 Y: 15 Cf.

Fukushima–Takeda [2; Th. 4.2.4] or Chen–Fukushima [1; Th. 2.1.11].

6.4

Donsker–Varadhan type large deviation principle

367

Lemma 6.4.6 below states that the function J./ D IE ./  2 ;  2 P ; enjoys the properties as a rate function in the large deviation principle. Lemma 6.4.6. The function J satisfies: (i) 0  J./  1. (ii) J is lower semicontinuous. (iii) For each ` < 1, the set ¹ 2 P W J./  `º is compact. (iv) J.02  m/ D 0 and J./ > 0 for  6D 02  m. We note that the identity J./ D IE 0 ./;

 2 P;

(6.4.29)

holds true, where IE 0 is defined in terms of the Dirichlet form (6.4.27) by ´

p p p E 0 . f ; f / if  D f 02  m; f 2 F 0 IE 0 ./ D 1 otherwise:

(6.4.30)

From Theorem 6.4.1 and Corollary 6.4.2, we obtain the next large deviation principle: e x;t º t >0 be a family of probability measures defined by Theorem 6.4.6. Let ¹Q e x;t º t >0 obeys the large deviation principle with (6.4.27). Then the sequence ¹Q rate function J : (i) For each open set G  P lim inf t !1

1 e x;t .G/   inf J./: log Q 2G t

(ii) For each closed set K  P lim sup t !1

1 e x;t .K/   inf J./: log Q 2K t

e x;t converges weakly to ı 2 as t ! 1. Corollary 6.4.4. The measure Q  m 0

Proof. If a closed set K does not contain 02  m, then inf2K J./ > 0 by e x;t .K/ D 0 Lemma 6.4.6 (iii), (iv). Hence Theorem 6.4.6 (ii) says that lim t !1 Q

368

6 Transformations of forms and processes

e x;t .K c / D 1. For a positive constant ı and a bounded continand lim t !1 Q uous function f on P , define the closed set K  P by K D ¹ 2 P W jf ./  f .02  m/j  ıº. Then we have ˇ Z ˇZ ˇ ˇ 2 ˇ e x;t .d/  f .0  m/ˇ  e x;t .d/ f ./Q jf ./  f .02  m/jQ ˇ ˇ P P Z Z 2 e e x;t .d/ jf ./  f .0  m/jQx;t .d/ C jf ./  f .02  m/jQ D Kc

K

e x;t .K/ C ı Q e x;t .K / ! ı  2kf k1 Q c

as t ! 1. Since ı is arbitrary, the proof of the corollary is complete. On account of Corollary 6.4.4, we can regard Theorem 6.4.6 as a genuine large deviation principle from the ground state.

Chapter 7

Construction of symmetric Markov processes

It is shown in this chapter that, given a regular Dirichlet space, there exists a Hunt process whose Dirichlet space is the given one. Such a process is unique up to an equivalence in accordance with Theorem 4.2.8. Furthermore, the local property of the given regular Dirichlet space is equivalent to the sample path continuity of the constructed Hunt process by virtue of Theorem 4.5.1. As is stated in Theorem A.2.2 of the Appendix, any transition function possessing the Feller property admits a Hunt process. In general, it is hopeless to construct a Feller transition function from the L2 -semigroup T t associated with the given Dirichlet space. However, if the Dirichlet space is regular, then the potential theory of Chapter 2 provides us with quasi continuous versions T t f and a sequence tn # 0 such that T tn f .x/ ! f .x/; n ! 1; q.e., for sufficiently many f . Going along a similar line as in the case of the Feller transition function, but ignoring successively the sets of capacity zero on which things might go wrong, we finally get a Hunt process outside some set of capacity zero.

A

7.1

e

Construction of a Markovian transition function

Let the pair .X; m/ be as in (1.1.7) and E be a regular Dirichlet form on L2 .XI m/. By making use of the potential theory developed in Chapter 2 in connection with the form E, we now construct an m-symmetric Hunt process on X whose Dirichlet form is the given one E. As a preliminary step, we construct in this section a transition function on X with time parameter QC . Here we denote by Q and QC the set of all rational numbers and positive rational numbers respectively. Lemma 7.1.1. Let F be a set. Consider a countable subset G of F and a countable collection S of mappings from F  F into F . Then there exists a countable set H such that (i) G  H  F , (ii) s.H  H /  H; 8s 2 S: Proof. Construct a sequence ¹s1 ; s2 ; : : :º such that each s 2 S appears infinitely often in the sequence. Let G0 D G; GnC1 D Gn [snC1 .Gn Gn /; n D 1; 2; : : : :

370

7 Construction of symmetric Markov processes

S Then the set H D 1 nD1 Gn is a desired one. In fact, for any x; y 2 H and s 2 S, we can find n such that x; y 2 Gn and s D snC1 . Let ¹T t ; t > 0º and ¹G˛ ; ˛ > 0º be the semigroup and resolvent on L2 .XI m/ associated with the form E (§1.4) and ¹eA ; A 2 O0 º be the 1-equilibrium potentials associated with E (§2.1). We use those notions and notations employed in Chapter 2. e1 (I) Markovian kernels ¹e p t ; t 2 QC º and R Lemma 7.1.2. There exist a regular nest ¹Fk0 º and Markovian kernels ¹e pt ; t 2 e QC º and R1 on X satisfying the following conditions: e1 .C1 .X//  C1 .¹F 0 º/, (i) e p t .C1 .X//  C1 .¹F 0 º/; R k

k

e1 u are quasi continuous version of T t u and G1 u respectively for (ii) e p t u and R any non-negative Borel u 2 L2 .XI m/. Proof. Let us first show that there exists a countable set B0  F \ C0 .X/ such that B0 is dense in C0 .X/ u; v 2 B0

a 2 Q ) juj 2 B0 ;

u C v 2 B0 ;

au 2 B0 :

In order to construct B0 , let ¹u1 ; u2 ; : : :º be a countable dense subfamily of C0 .X/. For each k and n, choose a uk;n 2 F \ C0 .X/ such that kuk  uk;n k1  1=n. The set G D ¹uk;n W k; n D 1; 2; : : :º is dense in C0 .X/. Define the mappings s0 ; s1 ; s2 ; : : : from ¹F \ C0 .X/º2 into F \ C0 .X/ by s0 .u; v/ D juj, s1 .u; v/ D u C v and si .u; v/ D ai u; i D 2; 3; : : : ; where ¹a2 ; a3 ; : : :º D Q. Applying the preceding lemma to G and S D ¹s0 ; s1 ; s2 ; : : :º, we get a countable set B0 satisfying (7.1.1) and (7.1.2). Let  [  H0 D T t .B0 / [ G1 .B0 /: .7:1:3/ t 2QC

Then H0 is countable and H0  F in view of Lemma 1.3.3. Choose a quasi e 0 D ¹e uW continuous version e u (in the restricted sense) of each u 2 H0 and let H 0 u 2 H0 º. By Theorem 2.1.2 (i), there exists then a regular nest ¹Fk º on X such e 0  C1 .¹F 0 º/. that H Sk 0 Now let Y0 D 1 pt kD1 Fk and fix t 2 QC . We construct a Markovian kernel e e1 can be constructed in the same way. By on X satisfying (i) and (ii). The kernel R

7.1

Construction of a Markovian transition function

371

Theorem 2.1.2 (ii),

B e e x 2 Y ; u; v 2 B ; B T .au/.x/ D aTe u.x/; x 2 Y ; u 2 B ; a 2 Q:

T t .uCv/.x/ D T t u.x/ C T t v.x/; t

t

0

0

0

0

(7.1.4) (7.1.5)

By the Markovian nature of T t , we also have 0  u  1;

e

u 2 B0 ) 0  T t u.x/  1;

8x 2 Y0 :

.7:1:6/

(7.1.4), (7.1.5) and (7.1.6) imply

e

jT t u.x/j  kuk1 ;

x 2 Y0 ; u 2 B0 :

.7:1:7/

It is then easy to see that, for each x 2 Y0 , there exists a unique positive linear functional lx on C1 .X/ satisfying

e

lx .u/ D T t u.x/;

8u 2 B0 ;

jlx .u/j  kuk1 ;

8u 2 C1 .X/:

.7:1:8/

To see this, choose for any u 2 C1 .X/ a sequence un 2 B0 such that kun  uk1 ! 0; n ! 1. Then T t un .x/ is uniformly convergent on Y0 . Denote the limit by lx .u/: lx is positive because lx .juj/ D limn!1 lx .jun j/  0. The linearity and the inequality in (7.1.8) are also clear. Note that lx .u/ belongs to the space C1 .¹Fk0 º/ as a function of x 2 Y0 because T t un 2 C1 .¹Fk0 º/. p t .x;  / on X such that lx .u/ D e p t u.x/ and lx admits a positive measure e e p t .x; X/  1; x 2 Y0 . Extend e p t by setting e p t .x; A/ D 0; 8x 2 X n Y0 ; 8A 2 B.X/. Then e p t becomes a Markovian kernel on X satisfying the condition (i). It now suffices to derive the property (ii) for u 2 C0C .X/ because then, by the same argument as in the proof of Theorem 4.2.3, we can get (ii) for non-negative u 2 L2 .XI m/. Choose un 2 B0 which is uniformly convergent to u 2 C0C .X/. Take w 2 B0 such that w  u. Then the functions vn D .0 _ un / ^ w belong to B0 and converge to u uniformly and in L2 -sense as well. Since e p t vn is a quasi continuous version of T t vn , a standard argument already adopted in the proof of Theorem 4.2.3 implies that e p t u is a quasi continuous version of T t u.

A

A

(II) A regular nest ¹Fk º Let ¹An º be a countable basis of open sets of X. We may assume that each An is relatively compact. Let O1 D ¹A W A is a finite union of An ’sº;

.7:1:9/

eA of then O1  O0 . For each A 2 O1 , choose a quasi continuous Borel version e the 1-equilibrium potential eA such that 0  e eA .x/  1; 8x 2 X.

372

7 Construction of symmetric Markov processes

e \ Bb .X/ such that e the smallest subfamily of F Denote by H e B0 ; ¹e (H.1) H eA ; A 2 O1 º, e/  H e ; t 2 QC ; R e1 .H e/  H e. (H.2) e p t .H e is an algebra over Q. (H.3) H e is then countable. To see this, consider the mappings from ¹F \ Bb .X/º2 H into F \ Bb .X/ defined by s1 W .u; v/ 7! u C v; s2;a W .u; v/ 7! au; s3 W e1 u. It then suffices to apply .u; v/ 7! uv; s4;t W .u; v/ 7! e p t u; s5 W .u; v/ 7! R Lemma 7.1.1 to G D B0 [ ¹e eA W A 2 O1 º and S D ¹s1 ; s2;a ; s3 ; s4;t ; s5 W a 2 Q, t 2 QC º. Lemma 7.1.3. There exists a regular nest ¹Fk º satisfying the following: If Y1 D S1 F kD1 k , then e  C1 .¹Fk º/; Fk  F 0 ; k D 1; 2; : : : ; (i) H k

(ii) e eA .x/ D 1; x 2 A \ Y1 ; A 2 O1 , e 1e e1 u.x/e tk R p tk u.x/ ! u.x/, .1=tk /.R p tk u.x// ! (iii) 9tk  QC , tk # 0, e e u.x/, x 2 Y1 , u 2 H , e, p s u.x/ D e p t Cs u.x/; x 2 Y1 ; t; s 2 QC ; u 2 H (iv) e pte e1 u.x/ D R e 1e e1 u.x/  R e1 u.x/; x 2 Y1 ; t 2 QC ; p t u.x/; e t e pt R (v) e pt R e C, u2H p te eA .x/  e eA .x/; x 2 Y1 ; t 2 QC ; A 2 O1 , (vi) e t e (vii) 0  e eA .x/  1; x 2 Y1 ; A 2 O1 , eB .x/; x 2 Y1 ; A; B 2 O1 ; A  B. (viii) e eA .x/  e Proof. By Lemma 2.1.4, e eA .x/ D 1 q.e. on A. Since O1 is countable, we can find a set N1 such that Cap.N1 / D 0 and e eA .x/ D 1;

8x 2 A n N1 ; 8A 2 O1 :

.7:1:10/

e admits a sequence tk 2 QC tk # 0 By Lemma 1.3.3 and Theorem 2.1.4, each u 2 H e such that (iii) holds for q.e. x 2 X. Since H is countable, ¹tk º can be selected e and (iii) holds for every u 2 H e and for every x 2 independently of u 2 H X n N2 ; N2 being a suitable set of zero capacity. By virtue of Theorem 2.1.2 (i), there exists a nest ¹Fk º satisfying the condiS tion (i) and X n 1 F kD1 k N1 [ N2 . Denote the m-regularization by ¹Fk º again. Obviously, (i), (ii), and (iii) are satisfied for this ¹Fk º. (iv) and (v) hold because they are valid m-a.e. (vi), (vii) and (viii) follow from Lemma 2.1.1 and Theorem 2.2.1.

7.2

Construction of a symmetric Hunt process

373

(III) A Markovian transition function p t ; t 2 QC Lemma 7.1.4. (i) There exists a Borel set Y2  Y1 with Cap.X n Y2 / D 0, e p t .x; X n Y2 / D 0; 8x 2 Y2 ; 8t 2 QC . (ii) Let

´ e p t .x; A/; p t .x; A/ D 0;

x 2 Y2 ; A 2 B.X/ x 2 X n Y2 ; A 2 B.X/;

.7:1:11/

Then ¹p t º t 2QC is a Markovian transition function on .X; B.X//; namely, each p t is a Markovian kernel on .X; B.X// and property .1:4:10/ is satisfied for every t; s 2 QC . p t .x; Proof. (i) T t 1X nY1 .x/ D 0 m-a.e. because X n Y1 is m-negligible. Since e X n Y1 / is a quasi continuous modification of T t 1X nY1 by Lemma 7.1.2 (ii), there .1/ .1/ exists a Borel sets Y1  Y1 such that Cap.X n Y1 / D 0; e p t .x; X n Y1 / D 0, .1/ 8x 2 Y1 ; 8t 2 QC . In this way, we can find a sequence of Borel sets Y1 .1/ .2/ .k/ .kC1/ Y1 Y1    such that e p t .x; X n Y1 / D 0; 8x 2 Y1 , 8t 2 QC , T .kC1/ .k/ 1 / D 0: Y2 D kD1 Y1 then possesses property (i). Cap.X n Y1 (ii) It suffices to show the semigroup property, p t ps u.x/ D p t Cs u.x/;

x 2 X; t; s 2 QC ; u 2 B0 :

.7:1:12/

Both sides vanish when x 2 X n Y2 . When x 2 Y2 , (i) implies p t ps u.x/ D pt . e p s u/.x/, which is also equal to e p t Cs u.x/ D p t Cs u.x/ by e p t .ps u/.x/ D e virtue of the preceding lemma.

7.2

Construction of a symmetric Hunt process

Let X D X [ ¹º be the one-point compactification of X . When X is already compact,  is adjoined as an isolated point. Extend the Markovian transition function p t on .X; B.X// of Lemma 7.1.4 to .X ; B.X // by 8